While the dissociative recombination (DR) of ground-state molecular ions with low-energy free electrons is generally known to be exothermic, it has been predicted to be endothermic for a class of transition-metal oxide ions. To understand this unusual case, the electron recombination of titanium oxide ions (TiO+) with electrons has been experimentally investigated using the Cryogenic Storage Ring. In its low radiation field, the TiO+ ions relax internally to low rotational excitation (≲100 K). Under controlled collision energies down to ∼2 meV within the merged electron and ion beam configuration, fragment imaging has been applied to determine the kinetic energy released to Ti and O neutral reaction products. Detailed analysis of the fragment imaging data considering the reactant and product excitation channels reveals an endothermicity for the TiO+ dissociative electron recombination of (+4 ± 10) meV. This result improves the accuracy of the energy balance by a factor of 7 compared to that found indirectly from hitherto known molecular properties. Conversely, the present endothermicity yields improved dissociation energy values for D0(TiO) = (6.824 ± 0.010) eV and D0(TiO+) = (6.832 ± 0.010) eV. All thermochemistry values were compared to new coupled-cluster calculations and found to be in good agreement. Moreover, absolute rate coefficients for the electron recombination of rotationally relaxed ions have been measured, yielding an upper limit of 1 × 10−7 cm3 s−1 for typical conditions of cold astrophysical media. Strong variation of the DR rate with the TiO+ internal excitation is predicted. Furthermore, potential energy curves for TiO+ and TiO have been calculated using a multi-reference configuration interaction method to constrain quantum-dynamical paths driving the observed TiO+ electron recombination.

## I. INTRODUCTION

^{1,2}Effectively a neutralization process, the DR reaction involves a free electron e

^{−}recombining with a positively charged molecular ion. For a diatomic species

*AB*

^{+}, this amounts to

where *A* and *B* symbolize the atomic fragments. As an exothermic non-radiative capture process, the DR reaction (1) can have a very high rate in cold gas-phase ionized media and by far dominates over the omnipresent radiative electron–ion recombination. Being a fundamental chemical process, DR has motivated a wide range of experimental and theoretical investigations. Often, these studies revealed intriguing beyond-Born–Oppenheimer dynamics behind the dissociation process, making the full understanding of DR a still challenging task.

A class of transition metal oxides (MOs) have been identified to potentially violate the exothermic DR behavior.^{3} For some MO cations, DR toward ground-state atomic fragments can become endothermic, implying that any dissociative electron capture process requires a net energy input into the system. Compared to the usual exothermic DR case, MO ions with endothermic DR are expected to much less efficiently neutralize with surrounding electrons.

*D*

_{0}(MO) > IE(MO), as further discussed in Sec. II B.

Previous studies based on chemical thermodynamic data^{3} identified a set of transition and lanthanide metal elements whose oxide ions MO^{+} potentially display the endothermic DR behavior. Notably, this group of elements includes titanium, which is also known to have a significant cosmic abundance.^{4} Thus, studying the DR of TiO^{+} on the one hand addresses endothermic DR as a fundamental process, while, on the other hand, it can also be expected to yield new evidence for the chemistry of titanium in cosmic environments.

The role of titanium atoms and titanium oxide molecules in astrophysical media has been discussed for the last five decades (e.g., Tsuji, Ref. 5). The TiO and TiO_{2} observations in hot stellar and circumstellar environments^{6} suggest that these molecules are essential in the dust grain formation process, acting as seeds for further silicate growth.^{7} In the colder interstellar medium (ISM), atomic titanium is known to be strongly depleted.^{8–10} Thus, also there, titanium oxide molecules, such as TiO, TiO_{2}, and, potentially, even TiO^{+}, are expected to behave as titanium storage. However, none of the searches for titanium oxide molecules in the ISM have been successful so far.^{11–13}

^{+}is preferred and the correspondingly endothermic DR does not allow the molecular ion to be destroyed in collisions with low-energy electrons. In the ISM, the CI reaction then would deplete Ti atoms and represent a source of free electrons.

^{14}By this, it would influence the ionization level and the electron-driven chemistry of other ions. In the opposite case of DR being exothermic, the same reaction would destroy TiO

^{+}ions and preferentially form atomic titanium. Hence, the depletion of Ti atoms observed in the ISM would need to be explained by other mechanisms, such as the formation of neutral titanium oxide molecules or the storage of atomic titanium in interstellar dust grains.

^{11}

As discussed further in Sec. II B, the pre-existing molecular and atomic data suggest that reaction (3) is close to thermoneutral, within an uncertainty of several tens of millielectronvolt (few hundred kelvin equivalent). More precise data are thus needed to finally resolve the question of CI exothermicity or, reciprocally, DR endothermicity. The new data may greatly improve our understanding on the titanium ISM chemistry.

To the best of our knowledge, neither the CI of Ti + O nor the DR of TiO^{+} has been investigated with respect to their reaction dynamics in previous experimental or theoretical work. The understanding of the corresponding energy balance is currently limited by the accuracy of TiO^{+} and TiO dissociation energies derived from room-temperature chemical reaction studies (Sec. II B). In this type of study, better energy resolution is hindered by the broad rotational excitation distribution of the molecules.

In our study, we investigate the energy balance of TiO^{+} DR using the Cryogenic Storage Ring (CSR) at the Max Planck Institute for Nuclear Physics in Heidelberg, Germany. In the low-temperature radiation field of this setup, the TiO^{+} molecules first cool down to the lowest rotational levels. Then, the energy balance of TiO^{+} is probed by collisions with cold electrons at controlled collision energy while measuring the energy output of the reaction in the form of kinetic energy released to Ti and O fragmentation products. Additionally, we determine collision-energy dependent rate coefficients for the TiO^{+} DR at various levels of TiO^{+} internal excitation.

The remainder of this paper is structured as follows: In Sec. II, we discuss the energy balance of the TiO^{+} DR reaction with a special emphasis on the molecular structure of TiO^{+}. Section III describes the CSR facility and the applied experimental approach. Section IV discusses the data analysis and presents the derived TiO^{+} DR reaction energy and the reaction rate coefficients. New quantum chemical calculations on the energy balance in the TiO^{+} DR and on TiO potential energy curves are presented in Sec. V. Finally, in Sec. VI, the experimental and calculated results are compared and discussed in the context of molecular dynamics theory and astrochemistry.

## II. ENERGY BALANCE IN DR OF TiO^{+}

^{+}DR reaction can be written as

where $ETiO+$, *E*_{Ti}, and *E*_{O} are the internal excitation energies of the involved reactants and products; *E*_{c} is the electron–ion collision energy in the center-of-mass frame; *E*_{KER} is the kinetic energy released to the Ti and O products, i.e., the center-of-mass kinetic energy of the products in their final states; and Δ*E* is the reaction energy for the ground-state reactants. It is here defined as the DR endothermicity (strictly speaking, endoergicity), i.e., as the minimum collision energy needed for the reaction to proceed from the ground state TiO^{+} to produce Ti and O in their respective ground states. Thus, positive Δ*E* suppresses the DR of ground-state TiO^{+} ions with electrons in the limit of zero collision energy.

*E*can be derived from the measured relative kinetic energy

*E*

_{KER}of the products at a controlled collision energy

*E*

_{c}. However, possible internal excitations of the reactants and products must also be taken into account, which complicate the relation between Δ

*E*and

*E*

_{KER}, as can be seen from the energy conservation equation,

*E*

_{Ti}, and

*E*

_{O}in Eq. (5) can lead to ambiguities in the interpretation of

*E*

_{KER}. While for the atomic Ti and O products only a limited number of electronic levels can play a role, for the molecular reactant TiO

^{+}, also rovibrational levels need to be considered. Section II A discusses the relevant energy levels of TiO

^{+}, Ti, and O. The expected energy balance in the DR of TiO

^{+}as based on previous studies is then discussed in Sec. II B.

### A. TiO, TiO^{+}, Ti, and O energy structure

Figure 1 shows the essential energy structure of the TiO and TiO^{+} molecules (potential curves calculated by Miliordos and Mavridis, Ref. 15) and the energy levels of the Ti and O reaction products. Given the low, sub-eV energies involved in our experimental approach, the states most relevant to our study lie energetically close to the TiO^{+}(*X*^{2}Δ) ground state. This includes the electronic fine-structure splitting and ro-vibrationally excited states of TiO^{+} and also the electronic states of Ti and O. These are the asymptotic, dissociated energy levels of neutral TiO electronic potential curves, whose detailed correlations to the separated-atom levels, however, are beyond the scope of this work.

The TiO^{+}(*X*^{2}Δ) ground term features a fine-structure doublet splitting^{16} of (26.26 ± 0.05) meV [spin–orbit coupling constant *A* = (105.9 ± 0.2) cm^{−1}], leading to *X*^{2}Δ_{3/2} and *X*^{2}Δ_{5/2} states with projection quantum numbers of the total angular momentum along the internuclear axis of Ω = 3/2 and 5/2, respectively. Higher electronically excited states *A*^{2}Σ^{+}, *B*^{2}Π, and *a*^{4}Δ appear only above ∼1.4 eV, as listed in Table I. The vibrational level spacing for both *X*^{2}Δ_{3/2} and *X*^{2}Δ_{5/2} is^{16} $\u223c0.13$ eV.

Level index . | TiO^{+}
. | Ti . | O . | |||
---|---|---|---|---|---|---|

I_{κ} | Term | $ETiO+$ (eV) | Term | E_{Ti} (eV) | Term | E_{O} (eV) |

0 | X^{2}Δ_{3/2} | 0.0 | a^{3}F_{2} | 0.0 | ^{3}P_{2} | 0.0 |

1 | X^{2}Δ_{5/2} | 0.026 | a^{3}F_{3} | 0.021 | ^{3}P_{1} | 0.020 |

2 | A^{2}Σ^{+} | 1.392 | a^{3}F_{4} | 0.048 | ^{3}P_{0} | 0.028 |

3 | B^{2}Π | 1.913 | a^{5}F_{1} | 0.813 | ^{1}D_{2} | 1.967 |

4 | a^{4}Δ | 3.193a | a^{5}F_{2} | 0.818 | ^{1}S_{0} | 4.190 |

Level index . | TiO^{+}
. | Ti . | O . | |||
---|---|---|---|---|---|---|

I_{κ} | Term | $ETiO+$ (eV) | Term | E_{Ti} (eV) | Term | E_{O} (eV) |

0 | X^{2}Δ_{3/2} | 0.0 | a^{3}F_{2} | 0.0 | ^{3}P_{2} | 0.0 |

1 | X^{2}Δ_{5/2} | 0.026 | a^{3}F_{3} | 0.021 | ^{3}P_{1} | 0.020 |

2 | A^{2}Σ^{+} | 1.392 | a^{3}F_{4} | 0.048 | ^{3}P_{0} | 0.028 |

3 | B^{2}Π | 1.913 | a^{5}F_{1} | 0.813 | ^{1}D_{2} | 1.967 |

4 | a^{4}Δ | 3.193a | a^{5}F_{2} | 0.818 | ^{1}S_{0} | 4.190 |

^{a}

Calculated value.^{15}

^{16}of $B0(3/2)=0.0695$ meV (0.5602 cm

^{−1}) and $B0(5/2)=0.0704$ meV (0.5682 cm

^{−1}). The spin–orbit coupling is handled as Hund’s case (a) with the standard formula

*F*(

*J*, Ω). Here,

*J*is the total angular momentum quantum number with lowest allowed values of

*J*= Ω, Λ = 2 for the Δ ground term and Σ = −1/2 or +1/2 for Ω = 3/2 or 5/2, respectively. This yields $\u223c20$ rotational energy levels of the

*X*

^{2}Δ

_{3/2}term below the lowest level of the

*X*

^{2}Δ

_{5/2}term. The rotational spacings can be read from Fig. 2(b). The rovibrational lifetimes taking into account the fine-structure coupling are discussed in Appendix B.

Both atomic products of TiO^{+} DR display triplet fine-structure splitting in the ground terms O(^{3}P) and Ti(a^{3}F) with individual excitation energies of $<50$ meV. The lowest electronic product states are summarized in Table I. The total internal energy levels of both fragments, *E*_{Ti} + *E*_{O}, relevant for the *E*_{KER} analysis according to Eq. (5), are shown in Fig. 1.

### B. TiO^{+} DR endothermicity from pre-existing data

*E*of a DR reaction can be obtained indirectly from the bond dissociation energies

*D*

_{0}and the ionization energies (IEs) of the involved ground-state species. Here, we discuss two most straightforward thermochemical relations,

^{19}[

*D*

_{0}(TiO) = (6.87 ± 0.07) eV] and in the two-color laser photoionization and photoelectron study by Huang

*et al.*

^{16}[IE(TiO) = (6.819 80 ± 0.000 10) eV]. For the approach in Eq. (7b), the best values were obtained in the guided-ion-beam tandem-mass-spectrometry study on the Ti

^{+}+ CO reaction by Clemmer

*et al.*

^{20}[

*D*

_{0}(TiO

^{+}) = (6.88 ± 0.07) eV] and in the two-color resonance-ionization spectroscopy experiment of Matsuoka and Hasegawa

^{21}[IE(Ti) = (6.828 12 ± 0.000 01) eV]. Both approaches give the same result of

^{+}DR reaction is likely slightly endothermic, but is thermoneutral within the experimental uncertainties.

^{22}More theoretical and experimental data on the molecular structure parameters for titanium oxides were summarized by Pan

*et al.*

^{23}

## III. EXPERIMENTAL

To determine the reaction energy and the collision energy-dependent rate coefficient of the TiO^{+} DR reaction, we use the merged electron–ion beam technique employed in the electrostatic Cryogenic Storage Ring (CSR)^{24} facility at the Max Planck Institute for Nuclear Physics in Heidelberg, Germany. Our study partially follows the approach used in the past for other DR studies at the CSR, e.g., for HeH^{+}, CH^{+}, and OH^{+} ions.^{25–27} Therefore, only a brief description of the general experimental approach is given here, complemented by the specific details on the TiO^{+} DR project.

### A. Merged-beams DR measurements at CSR

We generated an ∼20 nA beam of TiO^{+} ions using a sputtering Penning ion source (see Appendix A). The beam was accelerated to an energy of *E*_{ion} = 280 keV, passed through a magnetic mass filter (set to a mass-to-charge ratio of *m*/*q* = 64 u/*e*), and then, about 10^{6} ions were injected into the CSR. Once in the storage ring [Fig. 2(a)], the TiO^{+} ion beam was circulated at a frequency of $\u223c26$ kHz around the $\u223c35$ m closed orbit of the ring for a defined storage time. Due to the cryo-pumping effect, the temperature of $<6$ K in the CSR chamber results in residual gas densities of only $\u223c103$ cm^{−3}, allowing for the TiO^{+} storage of up to $\u223c1600$ s (exponential beam lifetime $\u223c1000$ s). As discussed in more detail in Appendix B, the low radiation field in CSR allows the ions to radiatively decay closer to their electronic and rovibrational ground states. This reduction in the internal excitation energy spread is beneficial for the DR energy balance measurements (lowering of *E*_{KER} ambiguities). After the radiative cooling phase, which extended for up to 1500 s, the DR reaction was probed by merging the stored ion beam with a collinear electron beam. The neutral DR products (Ti and O) were collected by using a microchannel plate (MCP) detector in the cryogenic region. The entire procedure (an *injection cycle*) was repeated until sufficient statistical quality of the data was obtained.

The low energy electron beam for the DR measurements was achieved using the electron cooler device. The electrons were extracted from a Ga–As photocathode, and the resulting beam was accelerated and expanded in the magnetic field to achieve low energy spreads of *k*_{B}*T*_{⊥} ≈ 2 meV and *k*_{B}*T*_{‖} = 0.2–2.5 meV in the transverse and longitudinal directions, respectively. Here, *k*_{B} stands for the Boltzmann constant. Guided by a solenoidal magnetic field, the electron beam was co-linearly merged with the ion beam in an electron–ion interaction zone.

*E*

_{e}is given by the electrostatic potential of a drift tube in the interaction zone. At the matched electron and ion velocities, the electron energy was

*E*

_{e}=

*E*

_{0}≈ 2.4 eV. From this condition, the electron beam energy

*E*

_{e}can be detuned to achieve the desired center-of-mass electron–ion collision energy,

^{28}

^{,}

^{29}was not achieved for the TiO

^{+}measurements because of a too strong contribution from dispersive heating.

^{30}Nevertheless, projection of the neutral products from the DR beam onto the imaging detector showed a good overlap of the beams, as further discussed in Appendix D. Electron cooling for heavy ions was later achieved at CSR by introducing zero dispersion in the electron–ion interaction zone (achromat mode) as demonstrated, for example, in recent work

^{27}on the DR of OH

^{+}.

In the DR-probing phase of the injection cycle, the electron detuning energy *E*_{d} was swapped repeatedly between four values with a dwell time of few tens of milliseconds each. In the first step, *E*_{d} is set to 0 eV (i.e., *E*_{e} = *E*_{0}). In this step, the electron beam drag force guarantees that the ion beam energy is kept constant even though non-zero detuning energies are applied in the other steps.^{31} In the next step, the energy is detuned to a desired value for DR measurement *E*_{m}. Here, *E*_{m} varies between the swapping cycles so that all desired collision energies are covered. In the next step, *E*_{d} was fixed to a reference value *E*_{d} = *E*_{r} ≈ 0.2 eV. The neutral-fragment count rate during this step was used as a proxy for the relative ion beam intensity, as further explained in Appendix D. In the last step, the electron beam was switched off to collect background data from processes not induced by electrons (among others, residual gas collisions and the detector dark count rate). Within each injection cycle, the four-step sequence was repeated until the duration of the DR probing phase was covered. A 5 ms waiting time was added between each of the steps to allow the electron beam to stabilize after changing the acceleration conditions.

The neutral products (Ti and O) resulting from DR in the electron interaction zone and from the ion collisions with the residual gas were separated from the stored ion beam in the downstream electrostatic deflector of the storage ring lattice. The products propagated ballistically to a 120 mm-diameter MCP detector backed by a phosphor (P-screen) anode.^{32} Here, the fragment impact positions manifested themselves as rapidly decaying light spots on the P-screen. Optical readout by using a silicon photomultiplier was used to provide triggers for impact-event counting. From the event count rates, the DR rate coefficient was derived as further discussed in Sec. III C. The time separation of the fragment impacts from a single DR event was well below 1 *µ*s and was thus resolved as single trigger only. At the typical impact rate of several kHz, deadtime effects and mixing of DR events within a single trigger can be neglected.

A separate, $\u223c1$ kHz frame-rate camera was used to read out the impact positions of DR fragments event by event with a spatial accuracy of ∼0.1 mm. The distance between the fragments reflects the kinetic energy release *E*_{KER} as further discussed in Sec. III B.

### B. DR reaction-energy analysis

We aim to deduce the TiO^{+} DR reaction energy Δ*E* from the kinetic energy release *E*_{KER} using the energy balance of Eq. (5). In turn, *E*_{KER} is determined from the Ti and O product distances as measured at the MCP-based imaging detector. A large advantage of our experimental method is that the collision energy *E*_{c} occurring in Eq. (5) can be varied. Hence, also such DR channels that are energetically forbidden because of positive Δ*E* or because of the fragment excitation energies *E*_{Ti} and *E*_{O} can be observed. Correspondingly, *E*_{KER} can be tracked by controlled variations of *E*_{c}, realized by varying the beam detuning energy *E*_{d}.

*d*of the reaction products on the detector scales with

*E*

_{KER}as

^{33}

*m*

_{Ti}and

*m*

_{O}are the Ti and O masses, respectively,

*D*≫

*d*is the distance between the dissociation point and the detector, and

*θ*is the angle between the Ti–O dissociation axis and the ion beam axis. As the molecular dissociation in DR occurs on time scales shorter than the molecular rotation,

*θ*represents the orientation of the ion at the time of the collision with the electron (axial recoil approximation, e.g., Ref. 34) and is thus randomized by the disordered orientation of the ions in the storage ring. Additionally, the finite length of the electron–ion interaction zone implies that the fragment flight distance

*D*distributes from

*D*

_{min}to

*D*

_{max}(where

*D*

_{max}−

*D*

_{min}describes the relevant interaction zone length). The parameters

*θ*and

*D*are unknown for individual events, and thus, also

*E*

_{KER}cannot be derived from

*d*on an event-to-event basis.

*E*

_{KER}, of a uniform distribution of

*D*between

*D*

_{min}and

*D*

_{max}, and of an isotropic distribution of the dissociation angles (uniform cos

*θ*), an analytical form $f\u22a50(d;EKER)$ can be derived for the fragment distance distribution, as described by Amitay

*et al.*

^{33}and shown in Fig. 3(b). The single-channel function $f\u22a50(d;EKER)$ displays a characteristic peak at

*d*=

*d*

_{peak}. This fragment distance corresponds to those dissociation events occurring closest to the detector (

*D*=

*D*

_{min}) and oriented transversely to the ion beam axis (

*θ*=

*π*/2). In combination with Eq. (10),

*E*

_{KER}can in this case be found from the measured

*d*

_{peak}as

*d*

_{peak}does not change except for extreme cases of anisotropy.

^{33}We also note that in our experiment, the Ti and O fragments from the same dissociation event impacted the detector within ≪10 ns so that the impact time difference could not be reliably measured and used in the less ambiguous 3D fragment imaging technique.

^{35}

Measured transversal fragment distributions and their comparison to the analytical $f\u22a50(d;EKER)$ were often used in DR experiments^{2,33,36} to determine *E*_{KER} for single isolated fragmentation channels, branching ratios, internal ionic excitation, and angular fragment distributions. However, in the present case, a large number of fragment channels with densely spaced *E*_{KER} are expected to overlap. They represent, in particular, the nine possible DR channels corresponding to combinations of fine structure excitation in Ti and O (see Table I with *I*_{Ti} = 0…2 and *I*_{O} = 0…2) and to the fine-structure and rotational excitation of TiO^{+} remaining even at long storage times (see Appendix B). All these contributions superimpose on each other with different amplitudes that cannot be predicted theoretically. In this situation, it is still practical for a general picture to consider the peak positions *d*_{peak} of possibly contributing DR channels in relation to the measured fragment distance distributions $f\u22a5\u0303(d)$. However, in order to obtain information about the underlying reaction energy Δ*E*, a more elaborate analysis is required. We model experimental transverse fragment distributions as a superposition of all possible DR channels in a detailed approach described in Appendix C. With relative DR channel amplitudes and the reaction energy Δ*E* as parameters, this yields model distributions *f*_{⊥}(*d*; Δ*E*) used to derive Δ*E* in a least-squares fitting procedure. In particular, this procedure aims at obtaining the value of Δ*E* that best represents the ensemble of experimental distributions $f\u22a5\u0303(d)$ observed for a range of detuning energies *E*_{d} between the ion and electron beams.

For our experimental parameters, Eq. (11) can be numerically written as $EKER=dpeak2\xd74.821$ meV/mm^{2} using *D*_{min} = 3300 mm, *E*_{ion} = 280 keV, and the Ti and O isotopic masses. This yields *d*_{peak} = 4.55 mm (7.89 mm) for *E*_{KER} = 100 meV (300 meV). At the typical electron energy spread represented by *k*_{B}*T*_{⊥} ≈ 2 meV, well-defined variations of *E*_{c} can be envisaged in steps down to $\u223c10$ meV. Hence, observing corresponding variations of the fragment distance distributions on the mm scale potentially offers access to the reaction energy balance [Eq. (5)] with an accuracy down to a few meV.

An example of measured fragment distance distributions $f\u22a5\u0303(d)$ is given in Fig. 4 for velocity-matched beam conditions (*E*_{d} = 0) and several storage-time intervals with DR sampling. While the distributions clearly show the strong decrease in fragment distances as the ions internally deexcite with increasing storage time (further discussed in Sec. IV A), the data also reflect the limited position resolution of the detector and include minor background contributions.

The fragment positions have been acquired by using a CMOS camera in an optical configuration with $\u223c2.4$ pixels corresponding to 1 mm in the detector plane. Each of the impact-generated spots on the phosphor screen anode covered an area of $\u223c20$ square-pixels following a Gaussian-like 2D distribution. Using the light intensities from the individual pixels, the spot-center position is determined with a precision of $\u223c0.1$ mm. However, spots from impacts at low transverse distances can overlap and thus affect the spot-position evaluation procedure. We find that the $f\u22a5\u0303(d)$ distribution is undistorted only for distances at *d* ≥ *d*_{min} = 1.6 mm. Hence, we also limit the fitting by the model distribution to distances *d* ≥ *d*_{min}. In the plots of acquired $f\u22a5\u0303(d)$ (e.g., Fig. 4), we also include the data in the range of *d* = 1.2–1.6 mm. Here, only a small distortion is expected, while the overall distribution shape can still be used for qualitative discussion. With Eq. (11), the limit *d*_{peak} = *d*_{min} corresponds to *E*_{KER} ∼ 12 meV.

Acquired fragment-pair events originate not only from the DR of TiO^{+} but also contain a background from collisions of the ions with the residual gas, from dark counts of the detector, and from DR of the TiOH^{+} contaminant (see Appendix A). The first two background contributions can easily be corrected for by subtracting reference data acquired without electron beam. Data in all $f\u22a5\u0303(d)$ plots in this paper have been corrected in this way. The contribution from TiOH^{+} DR cannot be subtracted as easily. Instead, we include the TiOH^{+} contribution in the model fragment distance distribution *f*_{⊥}(*d*; Δ*E*) as a term $f\u22a5bg(d)$, as described in Appendix C.

### C. DR rate coefficient analysis

The TiO^{+} DR rate coefficient measurement has been performed by using the electron beam as a collision target for the stored TiO^{+} ions. By adjusting the duration of the radiative cooling phase prior to the DR probing phase used in all injection cycles, the DR rate coefficient could be determined for various levels of TiO^{+} internal excitation.

The experimentally determined DR rate coefficient as a function of the detuning energy *E*_{d} is given as the merged-beams rate coefficient *α*^{mb}(*E*_{d}), obtained from the ion beam current, electron beam density, detector counting efficiency, and further experimental parameters, as explained in Appendix D. The rate coefficient *α*^{mb} reflects the electron–ion collision energy distribution specific to the particular CSR experiment. We present *α*^{mb}(*E*_{d}) for different storage-time intervals, which represent the various TiO^{+} internal excitation conditions realized in our experiment.

We also converted each of these results *α*^{mb}(*E*_{d}) to a kinetic-temperature rate coefficient *α*^{k}(*T*^{k}), which assumes a Maxwell–Boltzmann distribution at variable kinetic temperature *T*^{k} for the collision energies. The conversion procedure is described in Appendix E. In the literature, the kinetic-temperature rate coefficient *α*^{k} is often denoted as “plasma rate coefficient.” Note that, in this procedure, the TiO^{+} internal excitation is independent of *T*^{k} and continues to be given by the excitation conditions in the CSR for the respective storage time window.

## IV. EXPERIMENTAL RESULTS

### A. TiO^{+} DR reaction energy

We acquired fragment distance distributions $f\u22a5\u0303(d)$ at various DR-probing storage-time windows, ranging from 0 to 1600 s. The data were acquired over several injection cycles for a set of detuning energies *E*_{d} = 0, 10, 20, 30, 40, 60, 90, 120, and 150 meV. The evolution of the distance distribution $f\u22a5\u0303(d)$ as a function of storage time is represented in Fig. 4 for *E*_{d} = 0 eV. For determining the reaction energy Δ*E*, using the complete set of *E*_{d} values, we aimed at high signal intensity in connection with a good degree of TiO^{+} internal relaxation and, therefore, focused on the storage-time window of 500–600 s. The results for $f\u22a5\u0303(d)$ in this storage time window and for the set of non-zero *E*_{d} are represented in Fig. 5. In all displayed datasets, non-DR contributions have been subtracted already so that the remaining signal represents the DR of TiO^{+}, partly contaminated by the signal from DR of TiOH^{+}.

The *E*_{d} = 0 eV data in Fig. 4 demonstrate the internal cooling of TiO^{+} ions when stored in the CSR. In the 0–20 s probing interval, the $f\u22a5\u0303(d)$ distribution peaks at *d* ≈ 6 mm with a decaying tail at even larger distances. As *d*_{peak} = 4.55 mm for *E*_{KER} = 100 meV from Eq. (11), this shows that, even though *E*_{d} = 0 eV, the fragment energies exceed 100 meV for the hot ions soon after their injection into the CSR. At later storage times, the fragment distances strongly decrease and the $f\u22a5\u0303(d)$ distribution peaks at distances well below *d* = 1.6 mm (*E*_{KER} < 12 meV for *d*_{peak} < 1.6 mm). Similarly, for all storage times $\u2265500$ s, only a decaying tail up to *d* ≲ 3 mm is found (*E*_{KER} = 43 meV for *d*_{peak} = 3 mm). As discussed in Appendix C, the remaining flat part of $f\u22a5\u0303(d)$ at *d* > 3 mm can be well assigned to the DR of TiOH^{+}. Hence, we conclude that for the TiO^{+} DR, a strong reduction of *E*_{KER} occurs after longer storage.

Following the energy balance equation [Eq. (5)], we rationalize this observation by the internal cooling of the TiO^{+} ions. This is also supported by the fact that the drop of the observed *E*_{KER} follows well the trend of predicted TiO^{+} internal excitation from our radiative cooling model [see Fig. 2(b) and Table III]. Alternatively, the lower *E*_{KER} at long storage times could originate from increased excitation of the product Ti and O atoms (see Table IV) for internally cold TiO^{+}. However, such a strong dependence of product channel branching ratios on internal excitation has never been observed theoretically or experimentally.

Considering the energy balance in Eq. (5), the internal relaxation implies a decrease of the term $ETiO+$, where an average $\u27e8ETiO+\u27e9=11\xb12$ meV is expected in the storage-time window of 500–600 s according to Appendix B. Hence, the observed strong storage-related decrease of *E*_{KER} at *E*_{d} = 0 eV suggests that the sum *E*_{Ti} + *E*_{O} + Δ*E* is not strongly negative and can hardly be smaller than −20 meV. In this balance, however, the relative contributions of DR channels with various values of *E*_{Ti} and *E*_{O} remain unknown. In particular, it cannot simply be assumed that the DR signal remaining at *E*_{d} = 0 eV and low $\u27e8ETiO+\u27e9$ is due to ground-state Ti and O products only. To probe the participation of excited Ti and O products, we analyze the measurements at non-zero *E*_{d}, where all these DR channels (at least for the nine combinations of the lowest Ti and O fine-structure levels) are expected to become gradually accessible. While the individual amplitudes of these channels still remain uncertain, it can be expected that the fragment distance distributions for higher *E*_{d} will to some degree reflect the participation of all possible fine-structure excited DR channels.

According to Appendix C, Table IV, the values of *E*_{Ti} + *E*_{O} span a range of 76 meV with an irregular pattern, roughly appearing in four groups of more closely spaced values. In *E*_{KER} from Eq. (5), this leads to similarly distributed values, where in addition the energies $ETiO+$ from the various rotational levels (*J*,Ω) of TiO^{+} will be added. In the full distribution *f*_{⊥}(*d*), the respective transverse distance distributions for the various *E*_{KER} will be summed up.

In view of the complex situation of unknown and *E*_{d}-dependent relative contributions of the individual DR channels, we will proceed in two steps. First, we allocate a range with the limits Δ*E*_{min} and Δ*E*_{max} in which Δ*E* can at most vary considering the experimental $f\u22a5\u0303(d)$. Here, we compare the observed fragment distance distributions to the peak positions *d*_{peak} expected for single-*E*_{KER} channels from Eq. (11). In a second step, we vary Δ*E* within this range to obtain modeled transverse distance distributions *f*_{⊥}(*d*, Δ*E*) [ Appendix C, Eq. (C1)] where the individual channel amplitudes are determined from independent least-squares fits of *f*_{⊥}(*d*, Δ*E*) to $f\u22a5\u0303(d)$ for the various *E*_{d} values probed. From the overall minimum of the mean squared deviations *χ*^{2} reached for the different Δ*E*, we extract a most likely value of Δ*E* and estimate its uncertainty.

In the first step, we compare the acquired $f\u22a5\u0303(d)$ with a *stick diagram* *f*_{SD}(*d*), which represents the values *d*_{peak} from Eq. (11). Here, *E*_{KER} is obtained from the energy balance equation [Eq. (5)] for each possible channel combination (*J*, Ω, *I*_{Ti}, *I*_{O}) and setting *E*_{c} = *E*_{d}. This yields *f*_{SD}(*d*) as a sum of *δ*-like-function peaks at *d* = *d*_{peak}, over all the channels, with finite peak amplitudes weighted by the rotational-level populations *p*_{J;Ω} from the radiative cooling model ( Appendix B). The relative weights between the different (Ω, *I*_{Ti}, *I*_{O}) channels are arbitrarily set to 1. The resulting *stick* pattern of *f*_{SD}(*d*) is finally positioned along *d* by setting the reaction energy Δ*E* in Eq. (5) and compared with experimental $f\u22a5\u0303(d)$ for different Δ*E*.

Figure 6 shows the comparison for two values of *E*_{d} (rows) and four values of Δ*E* (columns). The experimental $f\u22a5\u0303(d)$ used in the upper row (*E*_{d} = 10 meV) shows contributions from TiO^{+} DR up to $\u223c2.5$ mm. The stick diagrams for different Δ*E* show that, irrespective of the individual peak amplitudes in the stick diagram, such contributions exist only if the reaction energy is below 40 meV. As a conservative upper limit, we hence obtain Δ*E*_{max} = 40 meV.

Conversely, the experimental $f\u22a5\u0303(d)$ in the lower row of Fig. 6 (*E*_{d} = 150 meV) shows a peak position near 4.6 mm, showing that DR channels exist with *E*_{KER} down to the value corresponding to this *d*_{peak}. If the reaction energy would be smaller than −10 meV, DR channels with such a small *E*_{KER} would not exist, irrespective of the individual channel amplitudes. Hence, as a lower limit, we obtain Δ*E*_{min} = −10 meV. Similar behavior compatible with these upper and lower limits of Δ*E* is found for the data at the other *E*_{d}.

In the second step, we use the detailed transverse fragment distance model *f*_{⊥}(*d*) of Eq. (C1) to fit the experimental data $f\u22a5\u0303(d)$, varying the channel amplitudes of $AITi;IO;\Omega $ and the background amplitude *B*, i.e., all free parameters except of Δ*E*. Details of the fitting procedure are given in Appendix C 3. With Δ*E* at a fixed value between Δ*E*_{min} and Δ*E*_{max}, the mean squared deviations *χ*^{2}(Δ*E*, *E*_{d}) are combined to a reduced value $\chi red2(\Delta E)$ according to Eq. (C3). The results for a grid of Δ*E* values within the range allocated in the first step are shown in Fig. 7.

The overall $\chi red2(\Delta E)$ (red circle symbols) is seen to assume a minimum at Δ*E* ≈ +4 meV, where $\chi red2(\Delta E)\u22481.5$. Toward lower Δ*E*, the function $\chi red2(\Delta E)$ displays a smooth rise without any additional features. Toward higher reaction energies, a localized increase of $\chi red2(\Delta E)$ occurs at Δ*E* ≈ 15 meV, followed by a secondary minimum at Δ*E* ≈ 24 meV. In Fig. 7, we also include the reduced squared deviations from only the *E*_{d} = 10 meV dataset, $\chi red,10meV2(\Delta E)$ (blue triangle symbols). At this very small *E*_{d}, the model distributions *f*_{⊥}(*d*, Δ*E*) turn out to be particularly sensitive to Δ*E*, and for Δ*E* > 0, a good fit to $f\u22a5\u0303(d)$ (see also Fig. 12) is only possible when Δ*E* lies close to the overall minimum near 4 meV, while the model for Δ*E* near the secondary minimum (24 meV) clearly cannot explain the experimental data, as also reflected by the large values of $\chi red,10meV2(\Delta E)$ in this region. The specific function $\chi red,10meV2(\Delta E)$ shows another minimum for Δ*E* < 0. However, this minimum contradicts the data with higher *E*_{d}, taken into account in the full $\chi red2(\Delta E)$. In fact, at small *E*_{d}, the sensitivity of *f*_{⊥}(*d*, Δ*E*) to the specific values of Δ*E* is lost for Δ*E* < 0 since many combinations of channel amplitudes can explain the experimental shape in this case. We note that the $\chi red,Ed2(\Delta E)$ curves for *E*_{d} other than 10 meV display shallow broad structures in the discussed Δ*E* region and thus do not contradict our reasoning here.

*E*= +4 meV as the estimate explaining best the experimental fragment distance distributions at all

*E*

_{d}. The deviation of the minimal $\chi red2(\Delta E)$ from the value of 1 expected for purely statistical deviations indicates an influence of systematical effects, such as the angular anisotropy of the DR cross section and its dependence on the rotational quantum number

*J*, which are not accounted for in our model. Hence, for deriving the uncertainty of the result, we deviate from the purely statistical definition of the uncertainty range, which at the large number of degrees of freedom $(\u223c300)$ in the combined fits would be of the order of 1 meV only. Instead, we find the uncertainty limits from the values of Δ

*E*where the deviation of $\chi red2(\Delta E)$ from its purely statistical value of 1 doubles in size. This leads us to the result of

### B. TiO^{+} DR rate coefficient

We acquired the DR data for TiO^{+} by probing the ion beam by electrons at various storage time windows, thus accessing different levels of TiO^{+} internal excitation. For each storage time window, the merged-beams rate coefficient *α*^{mb} was derived as a function of detuning energy *E*_{d}, following Eq. (D2) derived in Appendix D. Averaged within DR-probing windows of 0–20 s, 60–120 s, 500–600 s, 1000–1100 s, and 1500–1600 s, respectively, the resulting rate coefficients *α*^{mb}(*E*_{d}) are shown in Fig. 8. One-sigma error bars reflecting the uncertainty by the counting statistics are included in Fig. 8, while the total systematic scaling uncertainty was determined to be 34%, as described in Appendix D.

The *α*^{mb} rate coefficient is dominated by DR of ^{48}Ti^{16}O^{+}, complemented by other minor components. At detuning energies *E*_{d} ≳ 6.9 eV (corresponding to the TiO^{+} binding energy^{20}), the dissociative excitation (DE) of TiO^{+} by electrons becomes energetically accessible, resulting in Ti^{+} and O fragments. The applied detection technique cannot distinguish between the O fragments, reaching the detector after a DE event and the pair of Ti and O fragments resulting from DR. Therefore, the rate coefficient increase at *E*_{d} ≳ 6.9 eV may be, at least in part, given by the DE. On the other hand, for most diatomic molecular ions, DR is enhanced at *E*_{d} ≳ 10 eV due to the opening of new pathways for the reaction via higher lying repulsive (unbound) neutral potential curves.^{1} While such potential curves, converging to ground-state products Ti(a^{3}F) + O(^{3}P), can indeed be expected, we are unaware of any experimental or theoretical data that would allow us to safely assign the rate at *E*_{d} > 10 eV to either DR or DE.

Another contribution to *α*^{mb} is expected to arise from the DR of TiOH^{+}. The stored ^{48}Ti^{16}O^{+} ion beam was partly contaminated by ^{47}Ti^{16}OH^{+} isobars ( Appendix A). In Appendix C 2, we show that TiO^{+} DR and TiOH^{+} DR can be almost completely distinguished from each other via their strongly different *E*_{KER} and, consequently, different transverse distances of the DR products on the detector. For analyzing the contribution from TiOH^{+} DR, we choose the 500–600 s dataset and the collision energy range of *E*_{d} ≤ 150 meV. At the given storage times, the TiO^{+} ions are strongly deexcited and the transverse distances of Ti and O fragments from TiO^{+} DR do not exceed *d* = 8 mm for all *E*_{d} ≤ 150 meV. On the other hand, the large *E*_{KER} available in the DR of TiOH^{+} results in two-fragment events covering the full area of the 120 mm-diameter detector. Thus, to estimate the relative TiOH^{+} DR signal contribution, we compare the rate of two-fragment imaging events with *d* > 8 mm to the total two-fragment signal. Additionally, we take into account the fact that part of the two-fragment events cannot be resolved due to overlapping spots on the P-screen of the detector (*d* ≲ 1.6 mm) or escape registration because of too large transverse distance, in which case the lighter O fragment can miss the active detector area. We also consider various shapes of the TiOH^{+} DR signal at *d* < 8 mm [$f\u22a5bg(d)$, discussed in Appendix C 2]. The uncertainty of these effects is propagated to the final estimate of the TiOH^{+} DR contribution, as displayed by the dark-gray bars in Fig. 8 for detuning energies *E*_{d} = 0, 30, 60, 90, 120, and 150 meV. In the 500–600 s window, the fraction of the TiOH^{+} DR signal within the total DR signal ranges from 14% to 26%. We note that the plotted TiOH^{+} data represent the absolute DR rate coefficient of TiOH^{+} scaled by the relative fraction of the TiOH^{+} ions in the stored beam ($<10$%; see Appendix A).

Based on the comparatively more complex structure of TiOH^{+} with smaller rotational and vibrational energy intervals, we assume that its radiative cooling rate is slow compared to that of TiO^{+}, especially at the $>500$ s storage times. Therefore, the TiOH^{+} DR rate estimate derived from the 500–600 s storage-time data should be valid also for the later storage times. To our best knowledge, theoretical or experimental data on the DR of TiOH^{+}, which we could use for comparison, are unavailable. In Fig. 8, we add a hypothetical merged-beams rate coefficient assuming cross section varying as $\sigma (Ec)\u221dEc\u22121$ (representing the most usual DR behavior for polyatomic ions^{1}) and convolve it according to Eq. (E1) to yield *α*^{mb}. This background model clearly shows a steeper collision-energy dependence than the estimated experimental background. A better match was reached by creating a background model with a $\sigma (Ec)\u221dEc\u22120.85$ cross section dependence, as also indicated in Fig. 8. A more detailed analysis TiOH^{+} background would go beyond the scope of this paper.

The *α*^{mb}(*E*_{d}) curves reveal strong signal variations, both with *E*_{d} and with the storage time. At low detuning energies (*E*_{d} ≲ 30 meV, especially for short storage times), the strong decay of the rate coefficient with the increasing energy is a usual DR behavior.^{1,2} Additionally, we here observe a strong decrease of the rate coefficient for long storage times, which can be attributed to the radiative deexcitation of TiO^{+} and to the endothermicity of the DR process. As further discussed in Sec. VI, at long storage times, the TiO^{+} DR channel becomes energetically inaccessible for an increasingly large fraction of TiO^{+} ions as they reach low-lying rotational levels.

For the use of our DR data in plasma applications at various kinetic temperatures *T*^{k}, we converted the measured merged-beams rate coefficient *α*^{mb}(*E*_{d}) to the kinetic temperature rate coefficient *α*^{k}(*T*^{k}) following the procedure given in Appendix E. Additionally, we subtracted the estimated contribution from the TiOH^{+} DR contaminant signal using the $\sigma \u221dEc\u22120.85$ cross section in the background model as discussed for *α*^{mb} in Fig. 8. The resulting *α*^{k}(*T*^{k}) curves for the individual storage time windows are plotted in Fig. 9 in a temperature range *T*^{k} = 10–10^{4} K. The statistical uncertainties were propagated from *α*^{mb} to *α*^{k} and are displayed by the colored shaded areas in Fig. 9. They peak at the lowest temperatures of *T*^{k} ≈ 10 K and do not exceed 33%. The systematic uncertainty is dominated by the absolute scaling uncertainty (34%) and by uncertainties from the longitudinal and transverse electron temperatures *T*_{‖} and *T*_{⊥}, respectively, which contribute via the deconvolution procedure of *α*^{mb} to the cross section (see Appendix E and references therein). The longitudinal temperature *k*_{B}*T*_{‖} varies with the laboratory-frame electron energy and ranges from $\u223c2.4$ meV for *E*_{d} = 0 eV to $\u223c0.2$ meV at *E*_{d} ∼ 10 eV. Its uncertainty is dominated by a residual voltage ripple on the cathode potential and propagates to *α*^{k} as up to 50%. For the transverse temperature, $kBT\u22a5=2.0\u22120.5+1.0$ meV is assumed, which propagates to *α*^{k} as a scaling factor of up to 33%. The systematic uncertainties added in quadrature are shown as the additional gray band around the 500–600 s curve in Fig. 9 and reach up to 65% at *T*^{k} ≈ 10 K, while dropping down to $\u223c35%$ for *T*^{k} ≳ 300 K. The systematic uncertainties for the other curves are similar and are therefore not shown in Fig. 9. Also not shown in Fig. 9 is the estimated uncertainty from subtracting the TiOH^{+} DR rate, which reaches at most 10% for the 1500–1600 s curve and is lower for the other storage times.

## V. QUANTUM CHEMICAL CALCULATIONS

We complement our measurements of the TiO^{+} reaction energy Δ*E* by state-of-the-art quantum chemical calculations. To derive the DR reaction energy Δ*E*, i.e., the energy difference between the lowest energy Ti(4s^{2}3d^{2}; ^{3}F_{2}) + O(^{3}P_{2}) fragments and the ground vibrational level (*v* = 0) of TiO^{+}(*X*^{2}Δ_{3/2}), we perform coupled-cluster calculations with single, double, and perturbatively connected triple excitations, CCSD(T), and apply a correction for spin–orbit splitting. The wavefunctions of both TiO^{+} (at equilibrium) and the atomic fragments (*M*_{L} = ±2; ^{3}A_{1} under the C_{2v} point group) are of single-reference nature justifying the use of CCSD(T). The correlation consistent basis sets cc-pVXZ (X = T, Q, 5)^{37,38} were employed to construct the atomic and molecular orbitals. A series of diffuse basis functions were added in the basis set of oxygen (aug-cc-pVXZ).^{39} The equilibrium energies *E*_{e}, distances *r*_{e}, and harmonic vibrational frequencies *ω*_{e} for all basis sets are listed in Table S1 of the supplementary material [equilibrium constants for TiO(*X*^{3}Δ) are also provided for completeness]. We also included the electron correlation from the sub-valence 3s^{2}3p^{6} electrons of titanium. These core–valence calculations are denoted as CV-CCSD(T), and the corresponding weighted core basis sets were used for titanium (cc-pwCVXZ).^{37} Scalar relativistic effects were then added with the second order Douglas–Kroll–Hess (DKH2) Hamiltonian. In these calculations, CV-CCSD(T)-DKH2, the appropriate cc-pwCVXZ-DK basis set^{37} was used for titanium combined with the uncontracted aug-cc-pVXZ sets for oxygen. The equilibrium energies and frequencies were then extrapolated to the complete basis set (CBS) limit, assuming exponential convergence. Finally, spin–orbit corrections were taken from experimental data. Specifically, the energy decrease from Ti(^{3}F) to Ti(^{3}F_{2}) and from O(^{3}P) to O(^{3}P_{2}) was estimated as the energy difference between the *M*_{J}-averaged energy values of Ti(^{3}F) and O(^{3}P) and their lowest energy *J* = 2 component. The energy values were taken from Ref. 40, and the final spin–orbit corrections are Δ*E*_{SO}(Ti; ^{3}F) = −0.0276 eV and Δ*E*_{SO}(O; ^{3}P) = −0.0097 eV. For TiO^{+}, we used^{41} Δ*E*_{SO}(TiO^{+}; *X*^{2}Δ) = −Δ*E*_{e}(*X*^{2}Δ_{5/2}-*X*^{2}Δ_{3/2})/2 = −0.0260 eV/2 = −0.0130 eV since the two Ω components have the same degeneracy. From these values, Δ*E* is lower than the energy difference neglecting spin–orbit interaction by 0.0243 eV. Similarly, from TiO spectroscopy,^{42} we find Δ*E*_{SO}(TiO; *X*^{3}Δ) = −Δ*E*_{e}(*X*^{3}Δ_{3}-*X*^{3}Δ_{1})/2 = −0.0244 eV/2 = −0.0122 eV, which we use to find the spin–orbit corrections for *D*_{0}(TiO).

Table II lists ionization energies (IE_{e} and IE_{0}) and binding energies (*D*_{e} and *D*_{0}) for TiO obtained from the energies and frequencies of Table S1, where *ω*_{e}/2 was used for the zero-point vibrational corrections for IE_{0} and *D*_{0}. The values of IE_{e} and IE_{0} are practically identical, considering theoretical uncertainties, differing by less than 0.004 eV at any level of theory. Our best IE_{0} value [CV-CCSD(T)-DKH2] is 6.829 eV, which within $\u223c0.01$ eV is consistent with the experimental value of (6.8198 ± 0.0001) eV^{16} and 0.014 eV away from the recent coupled-cluster calculated value^{23} of 6.815 eV (including corrections due to quadruple electronic excitations). Note that the relativistic corrections are necessary to get this high level of agreement. Plain CCSD(T) and CV-CCSD(T) give 6.629 and 6.749 eV, respectively, at the CBS limit. On the other hand, the electron correlation energy of the 3s^{2}3p^{6} electron of titanium is important for obtaining a very accurate binding energy. Plain CCSD(T) gives *D*_{0} equal to 6.677 eV, which increases by 0.173 to 6.850 eV at CV-CCSD(T) and drops to 6.827 eV when scalar relativistic effects are included. Finally, the calculated energy differences Δ*E* between Ti(^{3}F) + O(^{3}P) and TiO^{+}(*X*^{2}Δ; *v* = 0) fluctuate around 0.0 eV depending on the methods or basis sets used. Positive values mean that TiO^{+} is lower in energy. Small basis sets tend to favor the TiO^{+} level, while at the CBS limit going from CCSD(T) to CV-CCSD(T), Δ*E* increases (both positive values). On the other hand, going from CV-CCSD(T) to CV-CCSD(T)-DKH2, Δ*E* drops and turns negative. The spin–orbit corrections, CV-CCSD(T)-DKH2-SO, favor the Ti(^{3}F) + O(^{3}P) level, decreasing Δ*E* even further to −0.0263 eV. Similarly, spin–orbit corrections to be applied to *D*_{0}(TiO) are −0.0251 eV, and the result of 6.827 eV from Table II is changed to 6.802 eV. In Sec. VI, these values will be compared to the present experiment and to results from others. Within that discussion, the calculated quantities will be referred to as Δ*E*^{calc} = −26 meV and $D0calc=6.802$ eV.

. | X = T . | X = Q . | X = 5 . | CBS . |
---|---|---|---|---|

CCSD(T) | ||||

IE_{e}a | 6.614 | 6.624 | 6.626 | 6.627 |

IE_{0}b | 6.616 | 6.626 | 6.629 | 6.629 |

D_{e}(TiO)c | 6.554 | 6.672 | 6.716 | 6.740 |

D_{0}(TiO)d | 6.491 | 6.610 | 6.653 | 6.677 |

ΔEe | −0.1250 | −0.0166 | 0.0242 | 0.0482 |

CV-CCSD(T) | ||||

IE_{e}a | 6.749 | 6.749 | 6.748 | 6.746 |

IE_{0}b | 6.752 | 6.752 | 6.751 | 6.749 |

D_{e}(TiO)c | 6.716 | 6.844 | 6.892 | 6.914 |

D_{0}(TiO)d | 6.653 | 6.781 | 6.829 | 6.850 |

ΔEe | −0.0988 | 0.0283 | 0.0780 | 0.1014 |

CV-CCSD(T)-DKH2 | ||||

IE_{e}a | 6.828 | 6.828 | 6.827 | 6.825 |

IE_{0}b | 6.831 | 6.831 | 6.830 | 6.829 |

D_{e}(TiO)c | 6.687 | 6.814 | 6.864 | 6.890 |

D_{0}(TiO)d | 6.624 | 6.750 | 6.800 | 6.827 |

ΔEe | −0.2072 | −0.0809 | −0.0300 | −0.0020 |

CV-CCSD(T)-DKH2-SO | ||||

D_{0}(TiO)f | 6.599 | 6.725 | 6.775 | 6.802 |

ΔEg | −0.2235 | −0.1052 | −0.0543 | −0.0263 |

. | X = T . | X = Q . | X = 5 . | CBS . |
---|---|---|---|---|

CCSD(T) | ||||

IE_{e}a | 6.614 | 6.624 | 6.626 | 6.627 |

IE_{0}b | 6.616 | 6.626 | 6.629 | 6.629 |

D_{e}(TiO)c | 6.554 | 6.672 | 6.716 | 6.740 |

D_{0}(TiO)d | 6.491 | 6.610 | 6.653 | 6.677 |

ΔEe | −0.1250 | −0.0166 | 0.0242 | 0.0482 |

CV-CCSD(T) | ||||

IE_{e}a | 6.749 | 6.749 | 6.748 | 6.746 |

IE_{0}b | 6.752 | 6.752 | 6.751 | 6.749 |

D_{e}(TiO)c | 6.716 | 6.844 | 6.892 | 6.914 |

D_{0}(TiO)d | 6.653 | 6.781 | 6.829 | 6.850 |

ΔEe | −0.0988 | 0.0283 | 0.0780 | 0.1014 |

CV-CCSD(T)-DKH2 | ||||

IE_{e}a | 6.828 | 6.828 | 6.827 | 6.825 |

IE_{0}b | 6.831 | 6.831 | 6.830 | 6.829 |

D_{e}(TiO)c | 6.687 | 6.814 | 6.864 | 6.890 |

D_{0}(TiO)d | 6.624 | 6.750 | 6.800 | 6.827 |

ΔEe | −0.2072 | −0.0809 | −0.0300 | −0.0020 |

CV-CCSD(T)-DKH2-SO | ||||

D_{0}(TiO)f | 6.599 | 6.725 | 6.775 | 6.802 |

ΔEg | −0.2235 | −0.1052 | −0.0543 | −0.0263 |

^{a}

IE_{e} = *E*_{e}(TiO^{+}; *X*^{2}Δ) − *E*_{e}(TiO; *X*^{3}Δ).

^{b}

IE_{0} = *E*_{0}(TiO^{+}; *X*^{2}Δ) − *E*_{0}(TiO; *X*^{3}Δ); *E*_{0} = *E*_{e} + *ω*_{e}/2.

^{c}

*D*_{e}(TiO) = *E*_{e}(Ti;^{3}F) + *E*_{e}(O;^{3}P) − *E*_{e}(TiO; *X*^{3}Δ).

^{d}

*D*_{0}(TiO) = *E*_{e}(Ti;^{3}F) + *E*_{e}(O;^{3}P) − *E*_{0}(TiO; *X*^{3}Δ).

^{e}

Δ*E* = *E*_{e}(Ti;^{3}F) + *E*_{e}(O;^{3}P) − *E*_{0}(TiO^{+}; *X*^{2}Δ). Positive values indicate that TiO^{+} is lower in energy.

^{f}

*D*_{0}(TiO) = *E*_{e}(Ti;^{3}F) + *E*_{e}(O;^{3}P) − *E*_{0}(TiO; *X*^{3}Δ) − 0.0251.

^{g}

Δ*E* = *E*_{e}(Ti;^{3}F) + *E*_{e}(O;^{3}P) − *E*_{0}(TiO^{+}; *X*^{2}Δ) − 0.0243.

To identify the crossings between the potential energy curve (PEC) of TiO^{+}(*X*^{2}Δ) and the repulsive part of the TiO PECs, we performed multi-reference configuration interaction (MRCI) calculations with aug-cc-pVTZ basis sets. The PECs in the region of the crossings are shown in Fig. 10. The MRCI calculations are based on complete active space self-consistent field (CASSCF) wavefunctions with the active space composed of the 4s3d/_{Ti} 2p/_{O} orbitals. The valence 2s/_{O} orbital was not included in the active space because of convergence issues, but excitations from it to the virtual space were included at the MRCI level. For the DR reaction, the spin coupling TiO^{+}(*S* = 1/2; *X*^{2}Δ) + e^{−}(*S* = 1/2) generates only singlets and triplets, and thus, only singlet and triplet states of TiO are considered here. The calculations were done within the C_{2v} point group, and the TiO states were averaged as ^{1}A_{1}/^{1}A_{2} (^{1}Σ^{±},^{1}Δ,^{1}Γ), ^{1}B_{1}/^{1}B_{2} (^{1}Π,^{1}Φ,^{1}H), ^{3}A_{1}/^{3}A_{2} (^{3}Σ^{±},^{3}Δ,^{3}Γ), and ^{3}B_{1}/^{3}B_{2} (^{3}Γ,^{3}Φ,^{3}H) separately at the CASSCF level. The PEC of TiO^{+} was calculated at the state specific CASSCF + MRCI level. To account for the inconsistency between the TiO and TiO^{+} PECs (state-averaged vs state-specific), we shifted the PECs of TiO^{+} by 0.029 698 a.u. (0.808 eV) to match the experimental TiO/TiO^{+} energy difference (rounded value from Ref. 16 of 6.82 eV). Figure 10 also includes the energy and wavefunction of the vibrational ground state of TiO^{+}. The TiO ground state fragments Ti(4s^{2}3d^{2}; ^{3}F) + O(^{3}P) generate the ^{1,3}(Σ^{+}[2], Σ^{−}[1], Π[3], Δ[3], Φ[2], Γ[1]) states, and nearly all of them cross the ground state TiO^{+}(*v* = 0) PEC within the Franck–Condon region (see Fig. 10). It should be emphasized that the above-mentioned energy shift moves the crossings by only ∼−0.04 Å, and thus, the shifting does not change the main observation that the TiO PECs cross the TiO^{+} PEC in the equilibrium region.

## VI. DISCUSSION

Using the combined merged-beams and fragment-imaging techniques, we experimentally determined the reaction energy for the dissociative recombination of TiO^{+} of Δ*E* = (+4 ± 10) meV (positive Δ*E* indicates endothermic DR). The reaction is thus thermoneutral within the uncertainties, with a slight bias toward endothermicity. Our result is well compatible with the value of Δ*E*^{prev} = (+0.05 ± 0.07) eV obtained from previous experimental data (Sec. II B), while the uncertainty improved by a factor of 7. In turn, our new value for Δ*E* can be used to derive updated values for the dissociation energies of TiO and TiO^{+} by inverting Eqs. (7a) and (7b). We obtain *D*_{0}(TiO) = Δ*E* + IE(TiO) = (6.824 ± 0.010) eV and *D*_{0}(TiO^{+}) = Δ*E* + IE(Ti) = (6.832 ± 0.010) eV, respectively. Also here, the precision of the new dissociation energies improves by a factor of 7 compared to the best previous experimental values.^{19,20}

Additionally, we determined values for the reaction energy Δ*E*^{calc} and the TiO dissociation energy $D0calc(TiO)$ using state-of-the-art coupled-cluster calculations. Our theoretical result Δ*E*^{calc} = −26 meV differs from the experiment by 30 meV—a difference well within the scatter between the various levels of the theory (see Sec. V). Interestingly, previous coupled-cluster calculations on TiO and TiO^{+} by Pan *et al.*^{23} yield Δ*E*^{calc} = +33 meV. On the other hand, our calculated $D0calc(TiO)=6.802$ eV lies by 30 meV below the experimental result derived above. Other coupled-cluster calculations^{23} yield $D0other(TiO)=6.853$ eV.

Furthermore, we measured the TiO^{+} DR merged-beams rate coefficient *α*^{mb}(*E*_{d}), which represents the reaction rate as a function of the collision energy at the experiment-specific energy spread (Fig. 8). The rate coefficient displays a decrease by a factor of up to 30 as the TiO^{+} ions deexcite while stored in the CSR. This trend is strongest at small collision energies *E*_{d} ≲ 10 meV and continues even at the longest investigated storage times of 1500–1600 s. Given the observed near-thermoneutrality of TiO^{+} DR and the expected low TiO^{+} excitation (Table III), the strong reaction to the internal excitation implies that efficient pathways leading to DR become active as soon as the energetic threshold of the reaction is overcome.

T (s)
. | p_{3/2} (%)
. | $\u27e8ETiO+\u27e9$ (meV) . | $\u27e8ETiO+\u27e93/2$ (meV) . | $\u27e8ETiO+\u27e95/2$ (meV) . |
---|---|---|---|---|

0–20 | $55(15$) | $142(5833$) | $129(6632$) | $130(6733$) |

60–120 | $62(25$) | $43(128$) | $34(117$) | $33(117$) |

500–600 | $86(23$) | 11(2) | $7.3(1.51.3$) | $7.3(1.41.3$) |

1000–1100 | 95(1) | 5.4(1.1) | 4.1(0.7) | 4.0(0.7) |

1500–1600 | 98(1) | 3.5(0.7) | 3.0(0.5) | 2.8(0.5) |

T (s)
. | p_{3/2} (%)
. | $\u27e8ETiO+\u27e9$ (meV) . | $\u27e8ETiO+\u27e93/2$ (meV) . | $\u27e8ETiO+\u27e95/2$ (meV) . |
---|---|---|---|---|

0–20 | $55(15$) | $142(5833$) | $129(6632$) | $130(6733$) |

60–120 | $62(25$) | $43(128$) | $34(117$) | $33(117$) |

500–600 | $86(23$) | 11(2) | $7.3(1.51.3$) | $7.3(1.41.3$) |

1000–1100 | 95(1) | 5.4(1.1) | 4.1(0.7) | 4.0(0.7) |

1500–1600 | 98(1) | 3.5(0.7) | 3.0(0.5) | 2.8(0.5) |

To better understand this, we investigated the PECs of those TiO electronic states, which may be formed by the capture of thermal electrons (down to a few meV) on TiO^{+} and then dissociate to Ti + O reaction products. Generally, in a DR reaction, the electron is captured by the molecular ion to form a neutral molecule in an energy-resonant excited state above the ionization threshold that dissociates into neutral fragments. For the “usual” exothermic DR, the final states can be reached without a barrier even in the limit of the lowest collision energies *E*_{c}. For this case, various pathways have been described.^{1,2,36} In particular, if neutral dissociating states cross the ionic potential within the Franck–Condon region of the ion, electron capture into these states is possible without any sharp dependence on *E*_{c} or on the internal molecular excitation energy (“direct” DR). In contrast, sharp features as a function of *E*_{c} are expected from “indirect” DR, where resonant electron capture occurs into ro-vibrationally excited neutral Rydberg states, which then pre-dissociate by coupling with another, dissociating excited neutral state. Barrier-less (exothermic) DR may still be suppressed for excited final states.

Since the process is governed by the electrostatic interaction in the multi-electron system, the excited states relevant for the DR of TiO^{+}(*X*^{2}Δ, *v* = 0) ions are expected to be those with total spin *S* = 0 and 1, given the spin 1/2 of the free electron. Accordingly, our calculated singlet and triplet PECs for TiO are shown in Fig. 10 together with the TiO^{+} ground-state PEC. The neutral curves correlate with various levels of the atomic dissociation products within an energy band of $\u223c0.08$ eV, as shown in Fig. 1. A large number of relevant neutral PECs cross the TiO^{+} ground state within the Franck–Condon region and would give rise to direct DR channels without a sharp energetic dependence in the “usual” case of exothermic DR. In the case of TiO^{+}, however, minimum energies *δE* exist for reaching all or most of the correlated final states. In a complex manner, the exact *δE* for each channel will depend on the potentials at larger internuclear distance, including possible barriers in the adiabatic potentials, the individual final atomic levels, and also the internal excitation of the colliding TiO^{+} ion. Nevertheless, it is plausible that for each of the channels, the DR rate will turn on with the typical smooth energy dependence of a “direct” DR process once the corresponding threshold *δE* is reached. Below the threshold, high vibrational levels in any of the electronic states of neutral TiO may resonantly couple to the TiO^{+} + e^{−} collision channel—although these levels cannot dissociate to Ti + O. Instead, they can either re-autoionize to TiO^{+} + e^{−} or stabilize to a lower-lying neutral TiO level by photon emission (photon-stabilized recombination). Small typical rates of only $\u223c10\u221212$ cm^{3} s^{−1} are generally envisaged^{2} for processes of the latter type.

Experimentally, we do not observe any significant resonant features in our DR spectra. This indicates that indirect DR, driven by neutral Rydberg levels at *E*_{c} > *δE*, is small for this system. Moreover, consistent with their expected small rates, features from resonant radiative recombination are also not observed. Instead, the experimental data for *α*^{mb}(*E*_{d}) and their dependence on the ion storage time suggest the presence of “direct,” but slightly endothermic DR pathways whose thresholds become more and more effective as the storage time increases and the internal excitation of the TiO^{+} ions reduces.

We demonstrate this by a model for *α*^{mb}(*E*_{d}) that takes into account the rotational and fine structure excitation of the TiO^{+} ions (see Sec. II A and Appendix B). Since the relative contributions of fine-structure excited final Ti and O states (Fig. 1) are unknown, we resort to the strongly simplifying assumption that all recombination leads to the Ti and O ground states so that DR fully sets in when the sum of *E*_{c} and the TiO^{+} excitation energy $ETiO+(J,\Omega )$ exceeds the endothermicity Δ*E*. Then, neglecting also the relative differences between the DR cross sections of the various channels, we assume the rate coefficient to be proportional to the number of TiO^{+} rotational levels above energetic threshold. For each of these levels, we assume a DR cross section varying as ∝1/*E*_{c} for $Ec>\Delta E\u2212ETiO+(J,\Omega )$, as typical for direct DR.^{2} We then convolve these cross sections with the experimental energy distribution to obtain a model for *α*^{mb}(*E*_{d}) (see Appendix F for more details).

After a scaling to the experimental data at *E*_{d} > 0.1 eV, the resulting model is shown in Fig. 11 for the experimental storage times and an example DR endothermicity Δ*E* = +14 meV. At the lowest *E*_{d}, the model, similar to the experimental data (Fig. 8), shows a rapid decrease of the rate coefficient as a function of storage time. In particular, at later times, only the remaining population in the Ω = 5/2 branch still contributes to DR. As *E*_{d} rises, Ω = 3/2 levels of TiO^{+} also contribute, resulting in the peak at *E*_{d} ≈ 20 meV. At even higher *E*_{d}, the model *α*^{mb}(*E*_{d}) decreases, reflecting the ∝1/*E*_{c} dependence, while, independent of storage time, all TiO^{+} levels contribute. The experimental data for the 500–600 s time window, included in Fig. 11, have a similar DR rate at low *E*_{d} as obtained for the model at Δ*E* = +14 meV. The same model for Δ*E* = +4 meV (not plotted) would show significantly higher rates in the same storage time window. This comparison alone could be taken as an indication that, within the uncertainty range of the result from the KER analysis, the larger Δ*E* values are more likely. However, the comparison in Fig. 11 also reveals a more complicated energy dependence of *α*^{mb}(*E*_{d}) than predicted by the simple model, together with a shift in the intermediate peak up to *E*_{d} ∼ 0.06 eV. This likely reflects the more detailed variation of the DR cross section with energy and between the channels corresponding to the different neutral PECs, as well as the influence of fine-structure excited Ti + O product channels. Reaching a better match between the model and the experimental data by including more details in the DR model is beyond the scope of this work. However, even the presented simple model strongly suggests that the decrease of the rate at low energies is due to the rotational cooling of the TiO^{+} levels below the energetic threshold and that the low-energy TiO^{+} DR rate coefficient would further decrease for even colder TiO^{+} ions.

From our measured *α*^{mb}(*E*_{d}), we also derived kinetic-temperature rate coefficients *α*^{k}(*T*^{k}) for the TiO^{+} DR in cold ionized media (Fig. 9) pertaining to the TiO^{+} excitation conditions realized in our various storage time windows. Especially at low temperatures (*T*^{k} ≲ 1000 K), the kinetic rate coefficients *α*^{k}(*T*^{k}) quickly drop as the internal TiO^{+} excitation becomes lower, reflecting the trend in the *α*^{mb}(*E*_{d}) data. In the dataset for the longest storage times (1500–1600 s), the TiO^{+} ions are expected to have on average $\u223c3.5$ meV internal excitation, with 98% of the population in the eight lowest rotational levels of the *X*^{2}Δ_{3/2} ground electronic state and $\u223c2$% remaining in the *X*^{2}Δ_{5/2} levels with a radiative equilibrium not yet reached. In interstellar space (thin, cold ISM), molecules often de-excite down to equilibrium with the cosmic microwave background at $\u223c2.7$ K or remain at temperatures of the order of 10 K because of collisional heating processes. For TiO^{+} under such equilibrium conditions, the limits of *T*_{exc} = 10 K (2.7 K) would imply significant population in only few of the lowest rotational levels for *X*^{2}Δ_{3/2} and very small relative populations of roughly 10^{−13} (10^{−48}) in *X*^{2}Δ_{5/2}. Based on the strong influence of the internal excitation of TiO^{+} within our simplified model of the DR process, it can be expected that compared to the experimental data, the rate coefficient would further drop for such even lower excitation stages.

Altogether, we consider our kinetic rate coefficient *α*^{k}(*T*^{k}) for the longest storage times (1500–1600 s) as an *upper limit* for cold-ISM conditions, which amounts to *α*^{k} ≤ 1 × 10^{−7} cm^{3} s^{−1} for *T*^{k} = 10 K. To estimate the *lower limit*, we consider the largest endothermicity within our uncertainty range, Δ*E*_{max} = 14 meV, where only the *X*^{2}Δ_{5/2} levels could still contribute to DR at the low ISM temperatures. Assuming that the rate coefficient scales with the change of the *X*^{2}Δ_{5/2} level population, the kinetic rate coefficient *α*^{k}(*T*^{k}) at *T*^{k} = 10 K would reach down below *α*^{k} < 5 × 10^{−13} cm^{3} s^{−1} (10^{−47} cm^{3} s^{−1}) for the discussed cold ISM conditions at *T*_{exc} = 10 K (2.7 K). Hence, our results include the possibility that the rate coefficient for the recombination of TiO^{+} ions with electrons in the cold ISM at *T*^{k} = 10 K would instead of DR be dominated by the non-dissociative radiative recombination with a generally predicted level of $\u223c10\u221212$ cm^{3} s^{−1}. Nevertheless, with the increasing kinetic temperature, the DR channel is expected to rise rapidly and reach a level between 10^{−8} and 10^{−7} cm^{3} s^{−1} at *T*^{k} = 100 K, similar to our 1500–1600 s experimental data.

In summary, we demonstrated that, even for a very complex case, the combined merged-beams and fragment-imaging techniques applied on internally cold ions in a cryogenic storage ring can yield thermochemistry data with a 10 meV accuracy. The TiO^{+} recombination with electrons is found to be thermoneutral within this uncertainty. At the cold ($<10$ K) conditions in the interstellar medium, the reaction cannot proceed faster than $\u223c1\xd710\u22127$ cm^{3} s^{−1}—a rate already an order of magnitude lower than usual for most other diatomic molecular ions. In addition, the low-temperature DR rate coefficient of TiO^{+} may be even significantly lower and is expected to vary strongly with the internal excitation of these ions. If TiO^{+} could ever be detected in cold ISM, its abundance would likely be a highly sensitive probe for the TiO^{+} internal excitation and, hence, the balance between collisional excitation processes and radiative relaxation. In the future, we plan to apply the here presented experimental approach to further systems expected to feature endothermic molecular-ion recombination, such as ZrO^{+}.

## SUPPLEMENTARY MATERIAL

See the supplementary material for the list of newly calculated Ti, O, and TiO+ energies and equilibrium parameters.

## ACKNOWLEDGMENTS

Financial support by the Max Planck Society is acknowledged. A.K. was supported, in part, by the U.S. National Science Foundation Division of Astronomical Sciences Astronomy and Astrophysics Grants program under Grant No. AST-1907188. E.M. acknowledges the James E. Land endowment (Auburn University) and the U.S. National Science Foundation (Grant No. CHE-1940456) for their financial support. AFRL support under Air Force Office of Scientific Research (Grant No. AFOSR-22RVCOR009) is acknowledged. The views expressed are those of the authors and do not reflect the official guidance or position of the Department of the Air Force, the Department of Defense (DoD), or the U.S. government. The appearance of external hyperlinks does not constitute endorsement by the United States DoD of the linked websites, or the information, products, or services contained therein. The DoD does not exercise any editorial, security, or other control over the information you may find at these locations.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Naman Jain**: Data curation (equal); Formal analysis (equal); Software (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). **Ábel Kálosi**: Data curation (equal); Formal analysis (equal); Software (equal); Visualization (equal); Writing – review & editing (equal). **Felix Nuesslein**: Conceptualization (equal); Investigation (equal); Methodology (equal); Writing – review & editing (equal). **Daniel Paul**: Formal analysis (equal); Investigation (equal); Software (equal); Visualization (equal); Writing – review & editing (equal). **Patrick Wilhelm**: Investigation (equal); Software (equal); Writing – review & editing (equal). **Shaun G. Ard**: Formal analysis (equal); Methodology (equal); Supervision (equal); Writing – review & editing (equal). **Manfred Grieser**: Investigation (equal); Writing – review & editing (equal). **Robert von Hahn**: Investigation (equal); Methodology (equal); Project administration (equal). **Michael C. Heaven**: Formal analysis (equal); Software (equal); Writing – review & editing (equal). **Evangelos Miliordos**: Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). **Dominique Maffucci**: Formal analysis (equal); Methodology (equal); Writing – review & editing (equal). **Nicholas S. Shuman**: Formal analysis (equal); Methodology (equal); Supervision (equal); Writing – review & editing (equal). **Albert A. Viggiano**: Conceptualization (equal); Funding acquisition (equal); Investigation (equal); Project administration (equal); Supervision (equal); Writing – review & editing (equal). **Andreas Wolf**: Conceptualization (equal); Investigation (equal); Project administration (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). **Oldřich Novotný**: Conceptualization (equal); Investigation (equal); Methodology (equal); Project administration (equal); Software (equal); Supervision (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

### APPENDIX A: TiO^{+} PRODUCTION AND ION BEAM CONTAMINATION

The Penning-sputtering ion source has been operated with O_{2} and Ar gases and a Ti-dominated-alloy sputtering target. While the oxygen gas was isotopically pure ^{16}O_{2}, for the used titanium target, a natural isotope distribution is expected, i.e., 8.3% of ^{46}Ti, 7.7% of ^{47}Ti, 73.7% of ^{48}Ti, 5.4% of ^{49}Ti, and 5.2% of ^{50}Ti. To maximize the yield of TiO^{+}, the ^{48}Ti^{16}O^{+} isotopologue with mass *m* = 64 u was chosen. Given the electrostatic acceleration scheme and the momentum filtering of the magnetic deflectors in the injection beamline, other ions of the same mass-to-charge ratio *m*/*q* = 64 u/*e* would get stored in the CSR along with ^{48}Ti^{16}O^{+}. To determine possible contaminant ions, the mass selected primary ions in the injection beamline were fragmented by collisions in an Ar-gas-cell, and the fragments were further mass analyzed by using a second magnetic filter.

Two isobaric contaminants have been considered in the analysis of the fragment mass spectra, i.e., $O4+16$ and ^{47}Ti^{16}OH^{+}. For $O4+16$ fragmentation, the detection of an O_{2} fragment within the injection beamline could be used as a marker. Unfortunately, the corresponding *m*/*q* = 32 u/*e* fragment mass-to-charge ratio is masked by the doubly charged ^{48}Ti^{16}O^{++} molecular ion. To resolve the ambiguity, we also analyzed fragment mass spectra for primary ions of *m*/*q* = 63 and 62 u/*e*, corresponding to ^{47}Ti^{16}O^{+} and ^{46}Ti^{16}O^{+}, respectively. Although for these primary-ion masses no contamination by $O4+$ is possible, the doubly charged ^{47}Ti^{16}O^{++} and ^{46}Ti^{16}O^{++} have been observed again. Moreover, the abundances of the respective ^{A}Ti^{16}O^{++} isotopologues (*A* being the nucleon number of Ti) scaled well with the primary beam intensity and with other unique titanium markers, such as ^{A}Ti^{+++}. From this behavior, we estimate that the $O4+16$ contamination did not exceed 1% of the total ion beam intensity. Furthermore, the DR imaging analysis did not show any fragmentation pattern, suggesting $O4+16$ contamination.

The potential ^{47}Ti^{16}OH^{+} contamination was investigated by ^{47}Ti fragments and by corresponding isotopologues, i.e., ^{A−1}Ti fragments for ^{A−1}Ti^{16}OH^{+} and ^{A}Ti^{16}O^{+} primary ions. For primary masses *m*/*q* = 64 and 63 u/*e* (*A* = 48 and 47, respectively), the ^{A−1}Ti fragments were clearly visible and we are unaware of any other contaminant at the corresponding mass. This was verified by choosing primary mass *m*/*q* = 62 u/*e* (*A* = 46) where the ^{A−1}Ti isotope does not exist, and we indeed did not observe any contribution at *A* − 1 = 45 in the fragment mass spectra. We estimate the relative fraction of ^{47}Ti^{16}OH^{+} in the *m*/*q* = 64 u/*e* primary beam to be less than 10%, assuming that the fragmentation cross section for TiOH^{+} in the gas cell is at least as large as that for TiO^{+}. As discussed further in Appendix C 2 and Sec. IV B, using a specific fragmentation pattern in DR of TiOH^{+}, we could further characterize the TiOH^{+} contribution to the imaging and rate data, respectively.

### APPENDIX B: RADIATIVE RELAXATION OF TiO^{+} IN CSR

The TiO^{+} ions were produced in a hot sputtering Penning ion source, and excitation of their internal degrees of freedom is thus expected to reach several thousand kelvin.^{43} After $\u223c40$ *μ*s, the extracted ions reached the CSR, where they were first stored for a predefined amount of time to deexcite before the DR probing by electrons started. The ion cooling is dominated by radiative cascading to reach the equilibrium with the CSR radiation field, which can be approximated by two components:^{44} 99% corresponding to $\u223c6$ K (representing the cryogenic chamber temperature) and 1% corresponding to 300 K (representing radiation leaks from outer environments, e.g., through the injection beamline). In this environment, we model the radiative relaxation of TiO^{+} using the Einstein coefficients for radiative transitions between the various electronic, vibrational, and rotational levels. As a result, we obtain TiO^{+} internal excitation distributions for the storage times equivalent to the DR probing in the experiment. Similar models of molecular-ion radiative cooling in the CSR radiation field were successfully tested in the past against *in situ* rotational state population probing methods.^{44–46}

To the best of our knowledge, there are no published data on radiative lifetimes for transitions in TiO^{+}. Therefore, we either estimate the lifetimes from other similar molecular systems or we calculate the corresponding Einstein coefficients for spontaneous emission ourselves. The radiative lifetimes for electronic transitions in the neutral TiO molecule^{47} do not exceed 5 *µ*s, which is similar also for other diatomic systems.^{48} Therefore, in our cooling model, we neglect the higher lying TiO^{+} electronic states A^{2}Σ^{+}, B^{2}Π, and a^{4}Δ as they are expected to relax already during the travel from the ion source to the CSR.

To estimate the vibrational level lifetimes, we follow the approach of Amitay *et al.* (Ref. 49) for calculating rovibrational radiative lifetimes, while intentionally neglecting the fine-structure splitting of the ground state doublet TiO^{+}(*X*^{2}Δ). The level spacing was obtained from the spectroscopic TiO^{+} parameters as given in Sec. II A. The dipole function was calculated as *μ* = 6.3 D and *dμ*/*dR* = 6.2 × 10^{8} D/cm using a def2-TZVP basis set.^{50} The vibrational radiative lifetime calculation reveals that even the slowest transition *v* = 1 ⟶ 0 does not exceed 60 ms. The vibrationally excited TiO^{+} levels are thus depopulated in CSR quickly, long before they could affect the DR probing phase, and therefore can be safely omitted from further discussion on the TiO^{+} radiative cooling model.

Thus, for the CSR experimental conditions, only rotational and TiO^{+}(*X*^{2}Δ_{3/2;5/2}) fine structure transitions remain relevant in the TiO^{+} radiative cooling model. For the remaining states, we again base the calculation of the Einstein coefficients for spontaneous emission on the approach of Amitay *et al.* (Ref. 49), but extend it for the treatment on multiplet electronic states.^{51,52} We calculate the needed Hönl–London coefficients by considering the ^{2}Δ state of TiO^{+} as intermediate between Hund’s coupling cases (a) and (b).^{51} For transitions within the Ω-branches, the results are nearly identical to pure Hund’s case (a),^{51} while the mixed character of the rotational wavefunctions gives rise to a non-negligible transition probability between the Ω-branches. The sum of the electric dipole Einstein coefficients from a single upper Ω = 5/2 level to all possible lower Ω = 3/2 levels is only weakly dependent on the *J* value of the upper level, and the related Ω = 5/2 ⟶ 3/2 radiative lifetime is $<700$ s for all levels. We also examined the possible role of magnetic dipole transitions and found that their additional contribution to the transition rates is ∼30% of those from the electric dipole.

We have generated a radiative relaxation model using the Einstein coefficients for spontaneous emission, while also accounting for stimulated emission and absorption by the CSR blackbody radiation. For the initial rotational excitation, we have taken a Boltzmann distribution at a temperature of 3000 K. The resulting rotational populations for the Ω = 3/2 and 5/2 branches of the electronic ground state are given in Fig. 2(b) as averages over four different storage time windows. It can be seen that, even though the initial rotational population spans widely over splitting of the Ω branches, at later storage times the two branches separate due to fast rotational cooling within each branch. The result of the radiative cooling model for TiO^{+} is summarized in Table III, listing the population fraction *p*_{3/2} of the Ω = 3/2 branch and mean excitation energies $\u27e8ETiO+\u27e9$ averaged over various storage time windows. The uncertainties of the modeled populations were estimated from the systematic uncertainty of the employed dipole moment *μ* and initial internal temperature of the ions *T*_{ini} using extreme values of *μ* = 5.8–6.8 D and *T*_{ini} = 1000–5000 K as guide. We note that the radiative cooling model does not include additional state-changing effects, such as reactive depletion of states with high DR cross section and inelastic electron–ion collisions. While both processes may occur during the DR-probing phase of the measurement, we estimate that especially at long storage times the populations are dominantly given by the radiative cooling.

### APPENDIX C: FRAGMENT DISTANCE DISTRIBUTION MODEL

#### 1. General approach

For TiO^{+} DR, a large number of densely-spaced *E*_{KER} channels can potentially contribute to the fragment imaging data. This is caused by the unknown excitation populations of Ti and O products of TiO^{+} DR (see Table I) and by the remaining fine-structure and rotational excitation of TiO^{+} even at long storage times (see Appendix B). Additionally, due to the experiment-specific effects, the electron–ion collision energies follow a non-trivial, continuous distribution function *f*(*E*_{c}; *E*_{d}) even at fixed experimental conditions (fixed detuning energy *E*_{d}). To take these ambiguities in the energy balance equation [Eq. (5)] into account, we build a model transverse distance distribution *f*_{⊥}(*d*; Δ*E*) combining all the *E*_{KER} channels available at given reaction energy Δ*E*, while the individual *E*_{KER} sub-channels are still represented by the $f\u22a50(d;EKER)$ distance distributions, as defined in Sec. III B. The individual channels and their weighting follow these rules:

All available combinations of Ti and O product excitations (

*E*_{Ti}+*E*_{O}< 1 eV) are considered as independent channels with individual DR cross sections and thus also independent weighting factors.The fine structure excitation branches Ω = 3/2 and 5/2 are also considered as independent channels with individual weighting factors.

The $AITi;IO;\Omega $ factors represent the weights for the

*I*_{Ti}-th Ti-product channel, the*I*_{O}-th O-product channel, and the Ω fine structure branch. The corresponding combined excitation-balance energies*E*_{Ω}−*E*_{Ti}−*E*_{O}are listed in Table IV. Here,*E*_{Ω}is the fine-structure excitation energy of TiO^{+}(*X*^{2}Δ_{Ω}) with respect to the ground state.Rotational states

*J*within one Ω branch follow populations*p*_{J;Ω}resulting from the radiative cooling model ( Appendix B).The collision energy distribution

*f*_{c}(*E*_{c};*E*_{d}) is considered as a continuous function based on the electron beam temperatures*T*_{⊥}and*T*_{‖}and other electron beam parameters (see Appendix E and parameters in Sec. IV B).Background not originating from DR of TiO

^{+}is represented by an additional distribution function $f\u22a5bg(d)$ with an independent scaling factor*B*.

Combined excitation-balance energies for TiO^{+}, Ti, and O
. | |||
---|---|---|---|

Ω . | Ti-state (I_{Ti})
. | O-state (I_{O})
. | E_{Ω} − E_{Ti} − E_{O} (meV)
. |

5/2 | a^{3}F_{2} (0) | ^{3}P_{2} (0) | +26.3 |

5/2 | a^{3}F_{2} (0) | ^{3}P_{1} (1) | +6.7 |

5/2 | a^{3}F_{3} (1) | ^{3}P_{2} (0) | +5.3 |

3/2 | a^{3}F_{2} (0) | ^{3}P_{2} (0) | 0.0 |

5/2 | a^{3}F_{2} (0) | ^{3}P_{0} (2) | −1.8 |

5/2 | a^{3}F_{3} (1) | ^{3}P_{1} (1) | −14.3 |

3/2 | a^{3}F_{2} (0) | ^{3}P_{1} (1) | −19.6 |

3/2 | a^{3}F_{3} (1) | ^{3}P_{2} (0) | −21.0 |

5/2 | a^{3}F_{4} (2) | ^{3}P_{2} (0) | −21.7 |

5/2 | a^{3}F_{3} (1) | ^{3}P_{0} (2) | −22.8 |

3/2 | a^{3}F_{2} (0) | ^{3}P_{0} (2) | −28.1 |

3/2 | a^{3}F_{3} (1) | ^{3}P_{1} (1) | −40.6 |

5/2 | a^{3}F_{4} (2) | ^{3}P_{1} (1) | −41.3 |

3/2 | a^{3}F_{4} (2) | ^{3}P_{2} (0) | −48.0 |

3/2 | a^{3}F_{3} (1) | ^{3}P_{0} (2) | −49.1 |

5/2 | a^{3}F_{4} (2) | ^{3}P_{0} (2) | −49.8 |

3/2 | a^{3}F_{4} (2) | ^{3}P_{1} (1) | −67.6 |

3/2 | a^{3}F_{4} (2) | ^{3}P_{0} (2) | −76.1 |

5/2 | a^{5}F_{1} (3) | ^{3}P_{2} (0) | −786.7 |

Combined excitation-balance energies for TiO^{+}, Ti, and O
. | |||
---|---|---|---|

Ω . | Ti-state (I_{Ti})
. | O-state (I_{O})
. | E_{Ω} − E_{Ti} − E_{O} (meV)
. |

5/2 | a^{3}F_{2} (0) | ^{3}P_{2} (0) | +26.3 |

5/2 | a^{3}F_{2} (0) | ^{3}P_{1} (1) | +6.7 |

5/2 | a^{3}F_{3} (1) | ^{3}P_{2} (0) | +5.3 |

3/2 | a^{3}F_{2} (0) | ^{3}P_{2} (0) | 0.0 |

5/2 | a^{3}F_{2} (0) | ^{3}P_{0} (2) | −1.8 |

5/2 | a^{3}F_{3} (1) | ^{3}P_{1} (1) | −14.3 |

3/2 | a^{3}F_{2} (0) | ^{3}P_{1} (1) | −19.6 |

3/2 | a^{3}F_{3} (1) | ^{3}P_{2} (0) | −21.0 |

5/2 | a^{3}F_{4} (2) | ^{3}P_{2} (0) | −21.7 |

5/2 | a^{3}F_{3} (1) | ^{3}P_{0} (2) | −22.8 |

3/2 | a^{3}F_{2} (0) | ^{3}P_{0} (2) | −28.1 |

3/2 | a^{3}F_{3} (1) | ^{3}P_{1} (1) | −40.6 |

5/2 | a^{3}F_{4} (2) | ^{3}P_{1} (1) | −41.3 |

3/2 | a^{3}F_{4} (2) | ^{3}P_{2} (0) | −48.0 |

3/2 | a^{3}F_{3} (1) | ^{3}P_{0} (2) | −49.1 |

5/2 | a^{3}F_{4} (2) | ^{3}P_{0} (2) | −49.8 |

3/2 | a^{3}F_{4} (2) | ^{3}P_{1} (1) | −67.6 |

3/2 | a^{3}F_{4} (2) | ^{3}P_{0} (2) | −76.1 |

5/2 | a^{5}F_{1} (3) | ^{3}P_{2} (0) | −786.7 |

*f*

_{⊥}(

*d*) is then represented by

*E*

_{KER}is still given by Eq. (5). The numerical implementation of Eq. (C1) uses a Monte Carlo sampling of the various involved distributions. The approximation of the background distribution function $f\u22a5bg(d)$ will be further discussed below.

We note that the model as given by Eq. (C1) assumes equal DR cross sections for rotational levels within the same Ω branch. While this assumption may generally not be correct (see, e.g., DR of HeH^{+}, Ref. 25), treating the individual *J*-channels independently would make the fitting of the model *f*_{⊥}(*d*) to the data numerically unstable. The validity of neglecting the *J* dependence is also supported by our observation that the fitted value of Δ*E* is not too sensitive on varying the scaling of *J*-channels within the Ω-branch, i.e., on the rotational excitation distribution.

#### 2. TiOH^{+} contamination model

To describe the TiOH^{+} contribution in terms $f\u22a5bg(d)$ of the model distribution *f*_{⊥}(*d*) [Eq. (C1)], we first analyze the expected ranges of transverse fragment distances from TiO^{+} DR, based on the approximate reaction energy balance, and then compare it to the acquired data. For the relaxed ions at storage times $>500$ s, the TiO^{+} excitation energies stay below $ETiO+\u227235$ meV (Table III). Thus, from Eq. (5) and from the lower estimate on the reaction energy Δ*E* ≳ −20 meV (Sec. II B), we derive that the relative product kinetic energies do not exceed *E*_{KER} ≈ 45 meV at *E*_{d} = 0 eV and stay below *E*_{KER} ≈ 200 meV even for *E*_{d} = 150 meV. Correspondingly, at these detuning energies, the transverse distances do not exceed *d* = 4 and 8 mm, respectively. However, the acquired $f\u22a5\u0303(d)$ distributions for cold ions at *E*_{d} = 0 eV (Fig. 4, $>500$ s) display clear contribution at *d* > 4 mm. This tail displays similar behavior also at non-zero collision energies, i.e., the main component smoothly decays from *d* ≈ 4 mm up to *d* ≈ 22 mm, and then further extends as an even weaker, nearly constant contribution until the detector size limit. As these distances cannot originate from DR of TiO^{+}, we assign it to DR of TiOH^{+}.

^{+}DR can, in principle, proceed via multiple fragmentation channels,

*E*for the respective channels, i.e., the channel endothermicities, which have been derived from the TiOH

^{+}binding energies

^{53}

^{,}

*D*(Ti

^{+}–OH) = 4.9 eV,

*D*(TiO

^{+}–H) = 2.3 eV, the OH binding energy

^{54}

^{,}

*D*

_{0}(O–H) = 4.4 eV, the TiH binding energy

^{55}

^{,}

*D*

_{0}(Ti–H) = 2.1 eV, the TiO ionization energy

^{16}IE(TiO) = 6.82 eV, and the thermoneutrality of TiO

^{+}DR reaction (Sec. II B). The Δ

*E*values reveal that channels (C2c) and (C2d) are energetically inaccessible at the collision energies employed in the imaging measurements (

*E*

_{d}≤ 0.15 eV). On the other hand, the fragments from channels (C2a) and (C2b) can impact the detector plane at distances of up to

*d*≲ 24 and 120 mm, respectively. As mentioned above, we indeed observe an “edge” in the background data at

*d*≈ 22 mm, indicating the Ti + OH channel. The fragmentation to TiO + H then covers the whole active detection area. Clearly, the nearly flat distance distribution from the TiOH

^{+}DR differs strongly from the sharp, single-

*E*

_{KER}case $f\u22a50(d)$. This likely results from the large number of possible excitation levels in the molecular DR products (OH and TiO), and such behavior has been observed also in DR of other polyatomic molecular ions.

^{56,57}

In the model distance distribution *f*_{⊥}(*d*) [Eq. (C1)], the TiOH^{+} DR is represented by the term $Bf\u22a5bg(d)$, where $f\u22a5bg(d)$ is the same for all fitted data (various *E*_{d} and storage times), while the scaling parameter *B* is determined in each fit. The $f\u22a5bg(d)$ distribution is approximated by a linear function, which well represents the data at *d* = 8–15 mm. The real shape of $f\u22a5bg(d)$ at low distances with overlapping DR of TiO^{+} (especially for *d* < 4 mm) is unknown. However, it can be well expected that with decreasing *d*, the distribution has to eventually reach also zero amplitude, similarly as the $f\u22a50(d)$ distribution for single *E*_{KER}. Various shapes of $f\u22a5bg(d)$ in this region have been tested, and only negligible effects on the fitted Δ*E* result were found.

The TiOH^{+} contribution to the rate coefficient data is further discussed in Sec. IV B.

#### 3. Fragment distance distribution fitting

*E*

_{d}and storage time window), we fit the acquired fragment distance distributions $f\u22a5\u0303(d)$ by the model distribution

*f*

_{⊥}(

*d*) from Eq. (C1), employing the least squares method to minimize

*χ*

^{2}as defined below. In view of the time-consuming Monte Carlo generation of the collision energy distribution

*f*

_{c}within the model function, the fitting procedure is performed step-wise for individual chosen Δ

*E*values. For each of these Δ

*E*and each

*E*

_{d},

*χ*

^{2}is minimized by varying $AITi;IO;\Omega $ and

*B*as free fit parameters. As a result, we obtain

*χ*

^{2}(Δ

*E*,

*E*

_{d}). To provide a measure for the combined fit quality from all

*E*

_{d}datasets, we generate a combined reduced

*χ*

^{2}as

*E*

_{d}datasets. The mean square deviation

*χ*

^{2}(Δ

*E*,

*E*

_{d}) is defined in the usual way as

*i*th acquired data point of the distribution and

*f*

_{⊥}(Δ

*E*,

*E*

_{d},

*d*

_{i}) is the model distribution result for transverse distance

*d*

_{i}, while

*σ*

_{i}is the standard deviation of the

*i*th experimental datapoint obtained from the event counting statistics.

Finally, *N*_{NDF}(*E*_{d}) is the number of degrees of freedom relevant for the fit in the given *E*_{d} dataset, i.e., the difference between the number of fitted experimental datapoints and the number of free fit parameters. Both quantities entering *N*_{NDF}(*E*_{d}) vary as a function of *E*_{d} for two reasons: (i) The upper limit of the fragment distance fit range (and thus also the number of points) was kept as low as possible for each *E*_{d} so that mainly distances covered by available TiO^{+} channels were covered. Moreover, (ii) only energetically open channels from Table IV at *E*_{c} ≈ *E*_{d} for the given *E*_{d} were included, i.e., *E*_{KER} > 0 (keeping a fixed list of energetically open channels for each *E*_{d} dataset in all the fits for the various Δ*E*). Additionally, for numerical stability reasons, we merged channels with combined excitation-balance energies *E*_{Ω} − *E*_{Ti} − *E*_{O} lying close to each other. By applying an energy difference limit of $<3.2$ meV, the number of channels thus reduced from 18 to 9, and the number of free amplitude parameters $AITi;IO;\Omega $ reduced correspondingly. The resulting groups are indicated in Table IV by horizontal separators. For each such group of channels, the combined excitation-balance energy was replaced by the mean value from the contributing channels. The applied channel grouping is not expected to change the fit results beyond other systematic uncertainties entering the model. With these modifications, *N*_{NDF}(*E*_{d}) ranged from 16 to 52.

Since the DR rate coefficient peaks at *E*_{d} = 0 eV, the corresponding dataset was acquired with much higher statistical quality compared to the data from higher detuning energies, which can also be seen by comparing the left and right columns in Fig. 5. To reduce the resulting strong weight of the *E*_{d} = 0 eV dataset in the $\chi red2(\Delta E)$ function, we deliberately increased the statistical uncertainties *σ*_{i} for the *E*_{d} = 0 eV data by a factor of 4 before the summation in Eq. (C4).

In Sec. IV A, we discuss the usage of $\chi red2(\Delta E)$ to determine the experimental reaction energy Δ*E*. We also employ the quantity $\chi red,Ed2(\Delta E)=\chi 2(\Delta E,Ed)/NNDF(Ed)$ representing the reduced *χ*^{2} obtained for only one selected *E*_{d} dataset. Figure 12 shows an example of the model transverse distance distributions fitted to the experimental data at the 500–600 s storage time window and various detuning energies *E*_{d}.

### APPENDIX D: MERGED-BEAMS RATE COEFFICIENT

In the CSR experiment, we obtain the merged-beams rate coefficient *α*^{mb} as a function of detuning energy *E*_{d} from the detected count rates *R*_{m}, *R*_{r}, and *R*_{o} within the DR-probing phase in the measurement, reference, and electron-off steps, respectively. As explained in Sec. III A, the experimental settings are fixed for the reference and electron-off data, while the detuning energy *E*_{d} = *E*_{m} is varied in the measurement step.

*α*

^{mb}, the non-electron-induced background is first subtracted from the measurement step data as

*n*

_{e}(

*E*

_{d}) is the electron density in the measurement step,

*N*

_{i}is the number of ions stored in CSR, $l\u0302$ is the effective length of the electron–ion interaction zone,

*C*= (35.12 ± 0.05) m is the CSR ion orbit circumference,

*η*is the detector counting efficiency factor, and

*ζ*is the fraction of the ion beam transverse cross section overlapped by the electron beam.

The electron density *n*_{e}(*E*_{d}) spanned the range of (0.6–4.1) × 10^{5} cm^{−3}, varying with the used *E*_{d}. The densities were obtained from the measured electron current ($\u223c5.1$ *μ*A) and the measured electron beam profile ($\u223c10$ mm diameter).

The effective overlap length of $l\u0302=(0.86\xb10.01)$ m is closely related to the length of the drift tube in the interaction zone and represents the central part of the electron–ion interaction zone with collision energy *E*_{d} derived from Eq. (8). Outside of the drift tubes, the collision energies deviate. The detailed procedure for determining the $l\u0302$ value was described by Kálosi *et al.*^{44}

The detection efficiency *η* is given by the probability that any of the neutral DR fragments (Ti or O) yields a count on the detector. As the exothermicity of the TiO^{+} DR is very low, all fragments geometrically reach the detector area. The detection efficiency is then given by the limited counting efficiency of the MCP (pore-area fraction) of^{26} *p* = 0.593 ± 0.015. Taking into account the probabilities to detect one or both of the fragments, the total counting efficiency is *η* = 2*p*(1 − *p*) + *p*^{2} = 0.83 ± 0.02.

To determine the transverse electron–ion beam overlap factor *ζ*, we used Ti and O fragment positions on the detector and the CSR beam-envelope (“betatron”) functions to obtain the ion beam profile. The electron beam was approximated by a cylindrical profile with an effective measured diameter. The transverse overlap then yields *ζ* = 0.86 ± 0.05.

The number of stored ions *N*_{i} in CSR was measured using a capacitive current pickup,^{24} which was used in connection with intermediate bunching of the stored ion beam using radio frequency acceleration. The details on the procedure and on the cross-calibration by an external Faraday cup are given in Appendix C of Paul *et al.*^{26} The uncertainty of the *N*_{i} determination in the present project was ±30%. We note that the various datasets within one DR-probing time window were first combined on a relative scale by using the reference-step signal *R*_{r} − *R*_{o} as a proxy for the ion beam intensity that was independent of the residual-gas pressure. The combined *α*^{mb}(*E*_{d}) result was then put on the absolute scale by a dedicated ion current measurement.

The total uncertainty of the absolute scaling was ±34%, obtained by adding in quadrature the uncertainties of the individual parameters in Eq. (D2). This total systematic uncertainty is dominated by the particularly difficult determination of the ion number *N*_{i}.

### APPENDIX E: KINETIC-TEMPERATURE RATE COEFFICIENT

*α*

^{mb}(

*E*

_{d}) to the kinetic-temperature rate coefficient

*α*

^{k}(

*T*

^{k}), the corresponding collision energy distributions must be taken into account. In general, the rate coefficient is defined as

*σ*(

*E*

_{c}) is the DR cross section as a function of collision energy

*E*

_{c}, $V(Ec)=2Ec/me$ is the collision velocity obtained from the collision energy and the electron mass

*m*

_{e}, and

*f*

_{c}(

*E*

_{c}) is the collision energy distribution. The merged-beams energy distribution in the CSR experiment is given by the detuning energy

*E*

_{d}, the transverse and longitudinal electron beam temperatures (

*T*

_{⊥}and

*T*

_{‖}), the merged-beams geometry, and the electron acceleration or deceleration near the ends of the drift tube in the interaction zone. Using a Monte Carlo forward simulation procedure, the CSR merged-beams energy distribution was modeled and used to deconvolve the experimental

*α*

^{mb}(

*E*

_{d}) to yield the cross section

*σ*(

*E*

_{c}). For details, see the original method by Novotný

*et al.*

^{58}and the updated method with the CSR-specific energy distribution, as described by Paul

*et al.*

^{26}The obtained

*σ*(

*E*

_{c}) was then converted to

*α*

^{k}(

*T*

^{k}) by Eq. (E1), assuming a Maxwell–Boltzmann energy distribution

*f*

_{c}=

*f*

_{MB}(

*E*

_{c};

*T*

^{k}) for the temperature parameter

*T*

^{k}.

### APPENDIX F: SIMPLE DIRECT DR MODEL

In the simple “level-counting” model, representing the direct DR pathway, the following assumptions are applied:

For TiO

^{+}rotational levels lying below the DR energetic threshold, $ETiO+(J,\Omega )+Ec<\Delta E$ [following the formalism in Eq. (5)], the DR cross section is*σ*= 0.For energetically accessible levels, the cross section scales as

*σ*=*A*/*E*_{c}, with*A*being an arbitrary scaling factor, identical for all*J*. The 1/*E*_{c}cross section dependence is typical for a direct DR process.The Ti and O DR products are assumed to be in their respective ground states only, i.e., there is no need to overcome the additional energy threshold due to the DR product excitation.

The cross section of each TiO

^{+}rotational level is weighted by the population of the given level, as given by the storage-time dependent TiO^{+}radiative cooling model in Appendix B.For each detuning energy

*E*_{d}, the collision energy distribution is represented by*f*_{c}(*E*_{c};*E*_{d}), following the formalism in Eq. (C1).The model merged-beams rate coefficient $\alpha mdlmb$ is convolved from the cross section

*σ*as $\alpha mdlmb=\u27e8\sigma v\u27e9$, similar to Eq. (E1).

## REFERENCES

*Dissociative Recombination of Molecular Ions*

_{2}in VY Canis Majoris

^{+}and TiO

*d*transition metal diatomic oxides and their ions: ScO

^{±}, TiO

^{±}, CrO

^{±}, and MnO

^{±}

*Tables of Spectra of Hydrogen, Carbon, Nitrogen, and Oxygen Atoms and Ions*

_{2}

^{+}, Ti

^{+}, and V

^{+}with CO. MC

^{+}and MO

^{+}bond energies

We are aware of the fact that the results of two independent measurements of Δ*E*_{prev} can be combined to reduce the uncertainty, here, $\Delta Eprevboth=(+0.05\xb10.05)eV$.

*ab initio*predictions for the ionization energies, bond dissociation energies, and heats of formation of titanium oxides and their cations TiO

_{n}/$TiOn+$,

*n*= 1 and 2

^{+}

^{+}and its implications for diffuse cloud chemistry

^{+}and its implications for the cosmic ray ionization rate in diffuse clouds

Here, the reduced mass of the electron–ion collision system is well approximated by the electron mass.

In the discussed context, the “drag force” is the longitudinal part of the electron cooling force. For the specific conditions of the TiO^{+} project, the longitudinal force was strong enough to observe ion energy dragging. However, the transverse cooling could not have been achieved due to the above-mentioned dispersive heating.

^{+}: Cross section and final states

*d*elements Sc–Zn

^{+}from multiphoton ionization photoelectron spectroscopy

^{+}ions in a cryogenic storage ring

*Ab initio*oscillator strengths and lifetimes for low-lying electronic states

^{+}, OH

^{+}, and SH

^{+}

*a*or

*b*

_{3}O

^{+}with cold electrons

^{+}using an ion storage ring